Infinite groups with a finiteness conditions on non-abelian - - PowerPoint PPT Presentation
Infinite groups with a finiteness conditions on non-abelian subgroups Patrizia LONGOBARDI UNIVERSIT DEGLI STUDI DI SALERNO Gruppen und Topologische Gruppen Groups and Topological Groups Department of Sociology and Social Research Trento
Infinite groups with a finiteness conditions
Patrizia LONGOBARDI
UNIVERSITÀ DEGLI STUDI DI SALERNO
Gruppen und Topologische Gruppen Groups and Topological Groups Department of Sociology and Social Research Trento June 16 - 17, 2017
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G.
Main Problem
Obtain information about the structure of G by looking at properties concerning M.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G.
Main Problem
Obtain information about the structure of G by looking at properties concerning M.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a (possibly infinite) group and let M be a family of subgroups of G.
Main Problem
Find information about the structure of G assuming that M satisfies a finiteness condition.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a (possibly infinite) group. Example Let M = L(G) be the family of all subgroups of G. Then
L(G) is finite ⇔ G is finite. What can be said about the structure of G if L(G), ordered by inclusion, satisfies, for instance, the maximal condition or the minimal condition?
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a (possibly infinite) group. Example Let M = L(G) be the family of all subgroups of G. Then
L(G) is finite ⇔ G is finite. What can be said about the structure of G if L(G), ordered by inclusion, satisfies, for instance, the maximal condition or the minimal condition?
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a (possibly infinite) group. Example Let M = L(G) be the family of all subgroups of G. Then
L(G) is finite ⇔ G is finite. What can be said about the structure of G if L(G), ordered by inclusion, satisfies, for instance, the maximal condition or the minimal condition?
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Is it true, for instance, that G is finite if L(G), ordered by inclusion, satisfies the maximal condition or the minimal condition?
Remark
There exist infinite simple groups in which every proper non-trivial subgroup has order a fixed prime p, the so called Tarski monsters.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Is it true, for instance, that G is finite if L(G), ordered by inclusion, satisfies the maximal condition or the minimal condition?
Remark
There exist infinite simple groups in which every proper non-trivial subgroup has order a fixed prime p, the so called Tarski monsters.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Is it true, for instance, that G is finite if L(G), ordered by inclusion, satisfies the maximal condition or the minimal condition?
Remark
There exist infinite simple groups in which every proper non-trivial subgroup has order a fixed prime p, the so called Tarski monsters.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
its Applications, vol.70 Kluwer Academic Publishers, Dordrecht, 1989.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
its Applications, vol.70 Kluwer Academic Publishers, Dordrecht, 1989.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
its Applications, vol.70 Kluwer Academic Publishers, Dordrecht, 1989.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
What can be said about the structure of G if L(G), ordered by inclusion, satisfies the maximal condition or the minimal condition? There are many well-known classical results about classes of groups G with L(G) ∈ Max or L(G) ∈ Min.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
What can be said about the structure of G if L(G), ordered by inclusion, satisfies the maximal condition or the minimal condition? There are many well-known classical results about classes of groups G with L(G) ∈ Max or L(G) ∈ Min.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
What can be said about the structure of G if L(G), ordered by inclusion, satisfies the maximal condition or the minimal condition? There are many well-known classical results about classes of groups G with L(G) ∈ Max or L(G) ∈ Min.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let P be a group theoretical property.
Definitions
Denote by LP(G) the family of all subgroups H of G such that H has P and by Lnon−P(G) the family of subgroups H of G such that H does not have P.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let P be a group theoretical property.
Definitions
Denote by LP(G) the family of all subgroups H of G such that H has P and by Lnon−P(G) the family of subgroups H of G such that H does not have P.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let P be a group theoretical property.
Definitions
Denote by LP(G) the family of all subgroups H of G such that H has P and by Lnon−P(G) the family of subgroups H of G such that H does not have P.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
We are interested in particular in the family Lnon−P(G), since if Lnon−P(G) is small, then many subgroups of G have P.
Example
If Lnon−P(G) = {G}, then every proper subgroup of G has P. Groups G with finiteness conditions
for various properties P have been studied by many authors.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
We are interested in particular in the family Lnon−P(G), since if Lnon−P(G) is small, then many subgroups of G have P.
Example
If Lnon−P(G) = {G}, then every proper subgroup of G has P. Groups G with finiteness conditions
for various properties P have been studied by many authors.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
We are interested in particular in the family Lnon−P(G), since if Lnon−P(G) is small, then many subgroups of G have P.
Example
If Lnon−P(G) = {G}, then every proper subgroup of G has P. Groups G with finiteness conditions
for various properties P have been studied by many authors.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
If P = ab is the property to be abelian, then Lab(G) is finite ⇔ G is finite.
Remark
Tarski monsters and more generally minimal non-abelian groups are groups G with Lnon−ab(G) finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
If P = ab is the property to be abelian, then Lab(G) is finite ⇔ G is finite.
Remark
Tarski monsters and more generally minimal non-abelian groups are groups G with Lnon−ab(G) finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
If P = ab is the property to be abelian, then Lab(G) is finite ⇔ G is finite.
Remark
Tarski monsters and more generally minimal non-abelian groups are groups G with Lnon−ab(G) finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let P = ab be the property to be abelian. Groups G in which Lab(G), ordered by inclusion, has Max or Min have been firstly studied respectively by A.I. Mal’cev in 1956 and O.J. Schmidt in 1945.
A.I. Mal’cev, On certain classes of infinite soluble groups, Mat. Sb. 28
(1951), 567-588 (Russian), Amer. Math. Soc. Transl. (2) 2 (1956), 1-21.
O.J. Schmidt, Infinite soluble groups, Mat. Sb., 17(59) (1945), 145-162
(Russian).
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let P = ab be the property to be abelian. Groups G in which Lab(G), ordered by inclusion, has Max or Min have been firstly studied respectively by A.I. Mal’cev in 1956 and O.J. Schmidt in 1945.
A.I. Mal’cev, On certain classes of infinite soluble groups, Mat. Sb. 28
(1951), 567-588 (Russian), Amer. Math. Soc. Transl. (2) 2 (1956), 1-21.
O.J. Schmidt, Infinite soluble groups, Mat. Sb., 17(59) (1945), 145-162
(Russian).
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Groups G in which Lnon−ab(G) has Max have been studied by L.A. Kurdachenko and D.I. Zaicev in 1991.
L.A. Kurdachenko, D.I. Zaicev, Groups with the maximum condition for
non-abelian subgroups, Ukrain. Mat. Zh. 43 (1991), 925-930 (Russian), English transl., Ukrainian Math. J. 43 (1991), 863-868.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Groups G in which Lnon−ab(G) has Max have been studied by L.A. Kurdachenko and D.I. Zaicev in 1991.
L.A. Kurdachenko, D.I. Zaicev, Groups with the maximum condition for
non-abelian subgroups, Ukrain. Mat. Zh. 43 (1991), 925-930 (Russian), English transl., Ukrainian Math. J. 43 (1991), 863-868.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Groups G in which Lnon−ab(G) has Max have been studied by L.A. Kurdachenko and D.I. Zaicev in 1991.
L.A. Kurdachenko, D.I. Zaicev, Groups with the maximum condition for
non-abelian subgroups, Ukrain. Mat. Zh. 43 (1991), 925-930 (Russian), English transl., Ukrainian Math. J. 43 (1991), 863-868.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Groups in which Lnon−ab(G) has Min have been studied by S.N. Černikov in 1964 and 1967.
S.N. Černikov, Infinite groups with prescribed properties of their systems of
infinite subgroups, Dokl. Akad. Nauk SSSR, 159 (1964) 759-760 (Russian), Soviet Math. Dokl., 5 (1964) 1610-1611.
S.N. Černikov, Groups with given properties of systems of infinite subgroups,
715-731.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Groups in which Lnon−ab(G) has Min have been studied by S.N. Černikov in 1964 and 1967.
S.N. Černikov, Infinite groups with prescribed properties of their systems of
infinite subgroups, Dokl. Akad. Nauk SSSR, 159 (1964) 759-760 (Russian), Soviet Math. Dokl., 5 (1964) 1610-1611.
S.N. Černikov, Groups with given properties of systems of infinite subgroups,
715-731.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let N be the property to be nilpotent. Groups G in which LN(G) has Max have been studied by M.R. Dixon and L.A. Kurdachenko in 2001.
M.R. Dixon, L.A. Kurdachenko, Groups with the maximum condition on
non-nilpotent subgroups, J. Group Theory 4 (2001), 75-87.
M.R. Dixon, L.A. Kurdachenko, Locally nilpotent groups with the maximum
condition on non-nilpotent subgroups, Glasgow Math. J. 43 (2001), 85-102.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let N be the property to be nilpotent. Groups G in which LN(G) has Max have been studied by M.R. Dixon and L.A. Kurdachenko in 2001.
M.R. Dixon, L.A. Kurdachenko, Groups with the maximum condition on
non-nilpotent subgroups, J. Group Theory 4 (2001), 75-87.
M.R. Dixon, L.A. Kurdachenko, Locally nilpotent groups with the maximum
condition on non-nilpotent subgroups, Glasgow Math. J. 43 (2001), 85-102.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let N be the property to be nilpotent. Groups G in which LN(G) has Max have been studied by M.R. Dixon and L.A. Kurdachenko in 2001.
M.R. Dixon, L.A. Kurdachenko, Groups with the maximum condition on
non-nilpotent subgroups, J. Group Theory 4 (2001), 75-87.
M.R. Dixon, L.A. Kurdachenko, Locally nilpotent groups with the maximum
condition on non-nilpotent subgroups, Glasgow Math. J. 43 (2001), 85-102.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
L.A. Kurdachenko, P. L., M. Maj, I.Ya Subbotin Groups with finitely many types of non-isomorphic non-abelian subgroups submitted.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
L.A. Kurdachenko, P. L., M. Maj, I.Ya Subbotin Groups with finitely many types of non-isomorphic non-abelian subgroups submitted.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G.
Definition
Consider the equivalence relation in M given by H ≃ K, with H, K ∈ M. Call the isomorphic type ItypeM of M any set of representatives of all equivalence classes in M.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G.
Definition
Consider the equivalence relation in M given by H ≃ K, with H, K ∈ M. Call the isomorphic type ItypeM of M any set of representatives of all equivalence classes in M.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G.
Definition
Consider the equivalence relation in M given by H ≃ K, with H, K ∈ M. Call the isomorphic type ItypeM of M any set of representatives of all equivalence classes in M.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G.
Definition
Consider the equivalence relation in M given by H ≃ K, with H, K ∈ M. Call the isomorphic type ItypeM of M any set of representatives of all equivalence classes in M.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G. Definitions Consider the equivalence relation in M given by H ≃ K, with H, K ∈ M. Call the isomorphic type ItypeM of M any set of representatives of all equivalence classes in M.
We study groups G in which ItypeM is finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G. Definitions Consider the equivalence relation in M given by H ≃ K, with H, K ∈ M. Call the isomorphic type ItypeM of M any set of representatives of all equivalence classes in M.
We study groups G in which ItypeM is finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group. Remark If G is non-trivial, then G, {1} ∈ ItypeL(G). Thus
|ItypeL(G)| 2.
Proposition
|ItypeL(G)| = 2 ⇔ either |G| a prime or G infinite cyclic.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group. Remark If G is non-trivial, then G, {1} ∈ ItypeL(G). Thus
|ItypeL(G)| 2.
Proposition
|ItypeL(G)| = 2 ⇔ either |G| a prime or G infinite cyclic.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group. Remark If G is non-trivial, then G, {1} ∈ ItypeL(G). Thus
|ItypeL(G)| 2.
Proposition
|ItypeL(G)| = 2 ⇔ either |G| a prime or G infinite cyclic.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition
|ItypeL(G)| = 2 ⇔ either |G| a prime or G infinite cyclic.
ItypeL(G) = {{1}, G}. Conversely, assume |ItypeL(G)| = 2, so ItypeL(G) = {{1}, G}. Then, for any x ∈ G − {1}, we have < x > ≃ G. Therefore G is cyclic. If G is finite, then there is an element y ∈ G of order a prime and G ≃ < y >, as required. // Problem
What about ItypeLab(G)?
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition
|ItypeL(G)| = 2 ⇔ either |G| a prime or G infinite cyclic.
ItypeL(G) = {{1}, G}. Conversely, assume |ItypeL(G)| = 2, so ItypeL(G) = {{1}, G}. Then, for any x ∈ G − {1}, we have < x > ≃ G. Therefore G is cyclic. If G is finite, then there is an element y ∈ G of order a prime and G ≃ < y >, as required. // Problem
What about ItypeLab(G)?
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition
|ItypeL(G)| = 2 ⇔ either |G| a prime or G infinite cyclic.
ItypeL(G) = {{1}, G}. Conversely, assume |ItypeL(G)| = 2, so ItypeL(G) = {{1}, G}. Then, for any x ∈ G − {1}, we have < x > ≃ G. Therefore G is cyclic. If G is finite, then there is an element y ∈ G of order a prime and G ≃ < y >, as required. // Problem
What about ItypeLab(G)?
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
What about |ItypeLab(G)|?
Remark
Obviously, if G = {1}, then {1}, < x > ∈ ItypeLab(G), where x ∈ G − {1}. Therefore |ItypeLab(G)| 2. But Tarski monsters T have |ItypeLab(T)| = 2.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
What about |ItypeLab(G)|?
Remark
Obviously, if G = {1}, then {1}, < x > ∈ ItypeLab(G), where x ∈ G − {1}. Therefore |ItypeLab(G)| 2. But Tarski monsters T have |ItypeLab(T)| = 2.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
What about |ItypeLab(G)|?
Remark
Obviously, if G = {1}, then {1}, < x > ∈ ItypeLab(G), where x ∈ G − {1}. Therefore |ItypeLab(G)| 2. But Tarski monsters T have |ItypeLab(T)| = 2.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Tarski monsters T have |ItypeLab(T)| = 2.
Proposition
Let G be a locally soluble group. Then |ItypeLab(G)| = 2 ⇔ either |G| a prime or G infinite cyclic.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Tarski monsters T have |ItypeLab(T)| = 2.
Proposition
Let G be a locally soluble group. Then |ItypeLab(G)| = 2 ⇔ either |G| a prime or G infinite cyclic.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Remark
Using a result due to V.S. Charin it follows that if a group G is such that ItypeL(G) or ItypeLab(G) is finite, then every abelian subgroup of G is minimax.
Definition A group G is said to be minimax if it has a finite series whose factors satisfy Min or Max. V.S. Charin, On soluble groups of type A4, Mat. Sbornik 52 (1960), no. 3, 895-914.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Remark
Using a result due to V.S. Charin it follows that if a group G is such that ItypeL(G) or ItypeLab(G) is finite, then every abelian subgroup of G is minimax.
Definition A group G is said to be minimax if it has a finite series whose factors satisfy Min or Max. V.S. Charin, On soluble groups of type A4, Mat. Sbornik 52 (1960), no. 3, 895-914.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Remark
Using a result due to V.S. Charin it follows that if a group G is such that ItypeL(G) or ItypeLab(G) is finite, then every abelian subgroup of G is minimax.
Definition A group G is said to be minimax if it has a finite series whose factors satisfy Min or Max. V.S. Charin, On soluble groups of type A4, Mat. Sbornik 52 (1960), no. 3, 895-914.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
What about |ItypeLnon−ab(G)|? Groups G with |ItypeLnon−ab(G)| = 1 have been studied by H. Smith and J. Wiegold in 1997.
subgroups, Rend. Math. Univ. Padova 97 (1997), 7-16.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Problem
What about |ItypeLnon−ab(G)|? Groups G with |ItypeLnon−ab(G)| = 1 have been studied by H. Smith and J. Wiegold in 1997.
subgroups, Rend. Math. Univ. Padova 97 (1997), 7-16.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Groups G with |ItypeLnon−ab(G)| = 1 have been studied by H. Smith and J. Wiegold in 1997.
Among other results they proved: Theorem
Let G be a soluble group. If G is isomorphic to every non abelian subgroup, then G contains an abelian normal subgroup of prime index.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Jointly with L.A. Kurdachenko, M. Maj and I.Ya Subbotin we studied groups G such that
Remark
Because of Tarski monsters, we have to assume something about G.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Jointly with L.A. Kurdachenko, M. Maj and I.Ya Subbotin we studied groups G such that
Remark
Because of Tarski monsters, we have to assume something about G.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Jointly with L.A. Kurdachenko, M. Maj and I.Ya Subbotin we studied groups G such that
Remark
Because of Tarski monsters, we have to assume something about G.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let D∞ be the infinite dihedral group: D∞ = < a, b | b2 = 1, ab = a−1 >. Then |ItypeLnon−ab(D∞)| = 1.
More generally: Remark
If G is a finitely generated abelian-by-finite group, then ItypeLnon−ab(G) is finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let D∞ be the infinite dihedral group: D∞ = < a, b | b2 = 1, ab = a−1 >. Then |ItypeLnon−ab(D∞)| = 1.
More generally: Remark
If G is a finitely generated abelian-by-finite group, then ItypeLnon−ab(G) is finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let D∞ be the infinite dihedral group: D∞ = < a, b | b2 = 1, ab = a−1 >. Then |ItypeLnon−ab(D∞)| = 1.
More generally: Remark
If G is a finitely generated abelian-by-finite group, then ItypeLnon−ab(G) is finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let D∞ be the infinite dihedral group: D∞ = < a, b | b2 = 1, ab = a−1 >. Then |ItypeLnon−ab(D∞)| = 1.
More generally: Remark
If G is a finitely generated abelian-by-finite group, then ItypeLnon−ab(G) is finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let A = Drp∈PAp, where Ap = {1} is an abelian p-group, and let G = A ⋊ < b >, where b2 = 1, ab = a−1, for any a ∈ A. Then ItypeLnon−ab(G) is infinite.
For, Ap < b > is non-abelian for any p ∈ P and Ap < b > ≃ Aq < b > for any p = q.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let A = Drp∈PAp, where Ap = {1} is an abelian p-group, and let G = A ⋊ < b >, where b2 = 1, ab = a−1, for any a ∈ A. Then ItypeLnon−ab(G) is infinite.
For, Ap < b > is non-abelian for any p ∈ P and Ap < b > ≃ Aq < b > for any p = q.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let A = Drp∈PAp, where Ap = {1} is an abelian p-group, and let G = A ⋊ < b >, where b2 = 1, ab = a−1, for any a ∈ A. Then ItypeLnon−ab(G) is infinite.
For, Ap < b > is non-abelian for any p ∈ P and Ap < b > ≃ Aq < b > for any p = q.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let A = Drp∈PAp, where Ap = {1} is an abelian p-group, and let G = A ⋊ < b >, where b2 = 1, ab = a−1, for any a ∈ A. Then ItypeLnon−ab(G) is infinite.
For, Ap < b > is non-abelian for any p ∈ P and Ap < b > ≃ Aq < b > for any p = q.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Example
Let A = Drp∈PAp, where Ap = {1} is an abelian p-group, and let G = A ⋊ < b >, where b2 = 1, ab = a−1, for any a ∈ A. Then ItypeLnon−ab(G) is infinite.
For, Ap < b > is non-abelian for any p ∈ P and Ap < b > ≃ Aq < b > for any p = q.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Lemma 1
Let G be a group with ItypeLnon−ab(G) finite. If K is an infinite locally finite subgroup of G, then K is abelian.
finite non-abelian subgroup F. Then G has an ascending chain F = F0 ≤ F1 ≤ · · · ≤ Fn ≤ Fn+1 ≤ . . .
case, the subgroups Fn and Fm cannot be isomorphic for n, m ∈ N, n = m, and we obtain a contradiction. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Lemma 1
Let G be a group with ItypeLnon−ab(G) finite. If K is an infinite locally finite subgroup of G, then K is abelian.
finite non-abelian subgroup F. Then G has an ascending chain F = F0 ≤ F1 ≤ · · · ≤ Fn ≤ Fn+1 ≤ . . .
case, the subgroups Fn and Fm cannot be isomorphic for n, m ∈ N, n = m, and we obtain a contradiction. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Lemma 1
Let G be a group with ItypeLnon−ab(G) finite. If K is an infinite locally finite subgroup of G, then K is abelian.
finite non-abelian subgroup F. Then G has an ascending chain F = F0 ≤ F1 ≤ · · · ≤ Fn ≤ Fn+1 ≤ . . .
case, the subgroups Fn and Fm cannot be isomorphic for n, m ∈ N, n = m, and we obtain a contradiction. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Lemma 1
Let G be a group with ItypeLnon−ab(G) finite. If K is an infinite locally finite subgroup of G, then K is abelian.
finite non-abelian subgroup F. Then G has an ascending chain F = F0 ≤ F1 ≤ · · · ≤ Fn ≤ Fn+1 ≤ . . .
case, the subgroups Fn and Fm cannot be isomorphic for n, m ∈ N, n = m, and we obtain a contradiction. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Lemma 2
Let G be a group with ItypeLnon−ab(G) finite. If G is locally nilpotent, then G is nilpotent.
Then G includes a non-abelian finitely generated subgroup K. Suppose that G is not nilpotent. Then G has an ascending chain K = K0 ≤ K1 ≤ · · · ≤ Kn ≤ Kn+1 ≤ · · ·
n ∈ N. But in this case the subgroups Kn and Km cannot be isomorphic for n, m ∈ N, n = m. This contradiction shows that G must be nilpotent. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Lemma 2
Let G be a group with ItypeLnon−ab(G) finite. If G is locally nilpotent, then G is nilpotent.
Then G includes a non-abelian finitely generated subgroup K. Suppose that G is not nilpotent. Then G has an ascending chain K = K0 ≤ K1 ≤ · · · ≤ Kn ≤ Kn+1 ≤ · · ·
n ∈ N. But in this case the subgroups Kn and Km cannot be isomorphic for n, m ∈ N, n = m. This contradiction shows that G must be nilpotent. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Lemma 2
Let G be a group with ItypeLnon−ab(G) finite. If G is locally nilpotent, then G is nilpotent.
Then G includes a non-abelian finitely generated subgroup K. Suppose that G is not nilpotent. Then G has an ascending chain K = K0 ≤ K1 ≤ · · · ≤ Kn ≤ Kn+1 ≤ · · ·
n ∈ N. But in this case the subgroups Kn and Km cannot be isomorphic for n, m ∈ N, n = m. This contradiction shows that G must be nilpotent. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Lemma 2
Let G be a group with ItypeLnon−ab(G) finite. If G is locally nilpotent, then G is nilpotent.
Then G includes a non-abelian finitely generated subgroup K. Suppose that G is not nilpotent. Then G has an ascending chain K = K0 ≤ K1 ≤ · · · ≤ Kn ≤ Kn+1 ≤ · · ·
n ∈ N. But in this case the subgroups Kn and Km cannot be isomorphic for n, m ∈ N, n = m. This contradiction shows that G must be nilpotent. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition 1
Let G be a group in which Lnon−ab(G) is finite. Let A be an infinite abelian periodic subgroup of G. Then NG(A) = CG(A).
Sketch of the proof. Let x be an arbitrary element of NG(A). We have A = Drp∈Π(A)Ap, where Ap is a Sylow p-subgroup of A, p ∈ Π(A). Clearly every subgroup Ap is < x >-invariant. If Ap is infinite, then x ∈ CG(Ap). In particular, Ap ≤ FC(< A, x >). If Ap is finite, then again we have the inclusion Ap ≤ FC(< A, x >). It follows that A ≤ FC(< A, x >). In this case x ∈ CG(A). //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition 1
Let G be a group in which Lnon−ab(G) is finite. Let A be an infinite abelian periodic subgroup of G. Then NG(A) = CG(A).
Sketch of the proof. Let x be an arbitrary element of NG(A). We have A = Drp∈Π(A)Ap, where Ap is a Sylow p-subgroup of A, p ∈ Π(A). Clearly every subgroup Ap is < x >-invariant. If Ap is infinite, then x ∈ CG(Ap). In particular, Ap ≤ FC(< A, x >). If Ap is finite, then again we have the inclusion Ap ≤ FC(< A, x >). It follows that A ≤ FC(< A, x >). In this case x ∈ CG(A). //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition 1
Let G be a group in which Lnon−ab(G) is finite. Let A be an infinite abelian periodic subgroup of G. Then NG(A) = CG(A).
Sketch of the proof. Let x be an arbitrary element of NG(A). We have A = Drp∈Π(A)Ap, where Ap is a Sylow p-subgroup of A, p ∈ Π(A). Clearly every subgroup Ap is < x >-invariant. If Ap is infinite, then x ∈ CG(Ap). In particular, Ap ≤ FC(< A, x >). If Ap is finite, then again we have the inclusion Ap ≤ FC(< A, x >). It follows that A ≤ FC(< A, x >). In this case x ∈ CG(A). //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition 2
Let G be a group, in which Lnon−ab(G) is finite. Let A be an abelian torsion-free subgroup of G. If NG(A) = CG(A), then A is minimax.
Sketch of the proof. Since NG(A) = CG(A), we can choose an element x ∈ NG(A) \ CG(A). It is possible to see that A has finite 0-rank. Let {a1, · · · , an} be a maximal Z-independent subset of A. Then the subgroup Aj = < aj ><x> is minimax. It follows that xkj ∈ CG(Aj), 1 ≤ j ≤ n. Put k = k1 · · · kn, then xk ∈ CG(< a1, · · · , an >). Since A is torsion-free and A is the pure envelope of < a1, · · · , an >, we have C<x>(< a1, · · · , an >) = C<x>(A). Then we obtain that A is minimax. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition 2
Let G be a group, in which Lnon−ab(G) is finite. Let A be an abelian torsion-free subgroup of G. If NG(A) = CG(A), then A is minimax.
Sketch of the proof. Since NG(A) = CG(A), we can choose an element x ∈ NG(A) \ CG(A). It is possible to see that A has finite 0-rank. Let {a1, · · · , an} be a maximal Z-independent subset of A. Then the subgroup Aj = < aj ><x> is minimax. It follows that xkj ∈ CG(Aj), 1 ≤ j ≤ n. Put k = k1 · · · kn, then xk ∈ CG(< a1, · · · , an >). Since A is torsion-free and A is the pure envelope of < a1, · · · , an >, we have C<x>(< a1, · · · , an >) = C<x>(A). Then we obtain that A is minimax. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Proposition 2
Let G be a group, in which Lnon−ab(G) is finite. Let A be an abelian torsion-free subgroup of G. If NG(A) = CG(A), then A is minimax.
Sketch of the proof. Since NG(A) = CG(A), we can choose an element x ∈ NG(A) \ CG(A). It is possible to see that A has finite 0-rank. Let {a1, · · · , an} be a maximal Z-independent subset of A. Then the subgroup Aj = < aj ><x> is minimax. It follows that xkj ∈ CG(Aj), 1 ≤ j ≤ n. Put k = k1 · · · kn, then xk ∈ CG(< a1, · · · , an >). Since A is torsion-free and A is the pure envelope of < a1, · · · , an >, we have C<x>(< a1, · · · , an >) = C<x>(A). Then we obtain that A is minimax. //
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Definition
A group G is called radical if there exists an ascending series of G with locally nilpotent factors.
Examples Any locally nilpotent group is a radical group. Any soluble group is a radical group. Definition
A group G is called generalized radical if G has an ascending series whose factors are either locally nilpotent or locally finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Definition
A group G is called radical if there exists an ascending series of G with locally nilpotent factors.
Examples Any locally nilpotent group is a radical group. Any soluble group is a radical group. Definition
A group G is called generalized radical if G has an ascending series whose factors are either locally nilpotent or locally finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Definition
A group G is called radical if there exists an ascending series of G with locally nilpotent factors.
Examples Any locally nilpotent group is a radical group. Any soluble group is a radical group. Definition
A group G is called generalized radical if G has an ascending series whose factors are either locally nilpotent or locally finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Definition
A group G is called radical if there exists an ascending series of G with locally nilpotent factors.
Examples Any locally nilpotent group is a radical group. Any soluble group is a radical group. Definition
A group G is called generalized radical if G has an ascending series whose factors are either locally nilpotent or locally finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Theorem A
Let G be a non-abelian locally generalized radical group. If ItypeLnon−ab(G) is finite, then G is a minimax, abelian-by-finite group, with Tor(G) finite.
Definition Tor(G) is the maximal normal torsion subgroup of G.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Theorem A
Let G be a non-abelian locally generalized radical group. If ItypeLnon−ab(G) is finite, then G is a minimax, abelian-by-finite group, with Tor(G) finite.
Definition Tor(G) is the maximal normal torsion subgroup of G.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Theorem A
Let G be a non-abelian locally generalized radical group. If ItypeLnon−ab(G) is finite, then G is a minimax, abelian-by-finite group, with Tor(G) finite.
Definition Tor(G) is the maximal normal torsion subgroup of G.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Definition
A group G is called generalized coradical if there exists a descending series of G whose factors are either locally nilpotent or locally finite.
Theorem B
Let G be a non-abelian generalized coradical group. If ItypeLnon−ab(G) is finite, then G is a minimax, abelian-by-finite group with Tor(G) finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Definition
A group G is called generalized coradical if there exists a descending series of G whose factors are either locally nilpotent or locally finite.
Theorem B
Let G be a non-abelian generalized coradical group. If ItypeLnon−ab(G) is finite, then G is a minimax, abelian-by-finite group with Tor(G) finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Definition
A group G is called generalized coradical if there exists a descending series of G whose factors are either locally nilpotent or locally finite.
Theorem B
Let G be a non-abelian generalized coradical group. If ItypeLnon−ab(G) is finite, then G is a minimax, abelian-by-finite group with Tor(G) finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Remark
If G is a finitely generated abelian-by-finite group, then ItypeLnon−ab(G) is finite.
Remark
The converse of Theorem A or the converse of Theorem B does not hold.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Remark
If G is a finitely generated abelian-by-finite group, then ItypeLnon−ab(G) is finite.
Remark
The converse of Theorem A or the converse of Theorem B does not hold.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Remark
If G is a finitely generated abelian-by-finite group, then ItypeLnon−ab(G) is finite.
Remark
The converse of Theorem A or the converse of Theorem B does not hold.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G. Definitions Consider the equivalence relation in M given by H ≃ K, with H, K ∈ M. Call the isomorphic type ItypeM of M any set of representatives of all equivalence classes in M.
We study groups G in which ItypeM is finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group and let M be a family of subgroups of G. Definitions Consider the equivalence relation in M given by H ≃ K, with H, K ∈ M. Call the isomorphic type ItypeM of M any set of representatives of all equivalence classes in M.
We study groups G in which ItypeM is finite.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group, and let M is the family of the commutator subgroups of all subgroups of G : M = {H′ |H ∈ L(G)}. The problem to study the structure of the group G in which ItypeM is finite has been studied by F. de Giovanni and D.J.S. Robinson in 2005, as well as by M. Herzog, P. L., M. Maj and D.J.S. Robinson, H. Smith in a series of papers (2006, 2013, 2014).
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Let G be a group, and let M is the family of the commutator subgroups of all subgroups of G : M = {H′ |H ∈ L(G)}. The problem to study the structure of the group G in which ItypeM is finite has been studied by F. de Giovanni and D.J.S. Robinson in 2005, as well as by M. Herzog, P. L., M. Maj and D.J.S. Robinson, H. Smith in a series of papers (2006, 2013, 2014).
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
subgroups, J. London Math. Soc. 71 (2005), no. 2, 658-668.
Group Theory 2004, Contemp. Math. Amer. Math. Soc., Providence, RI 402 (2006), 181-192.
classes of derived subgroups, Glasgow Math. J. 55 (2013), no. 3, 655-668.
isomorphism classes of derived subgroups, Proc. of "Group Theory, Combinatorics, and Computing", Boca Raton-Florida, Contemp. Math. 611 (2014), 121-135.
isomorphism classes of derived subgroups, J. Algebra, 393 (2013), 102-119.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Now together with L.A. Kurdachenko and M. Maj, we are considering the family M of all non-normal subgroups of a group G, assuming that ItypeM is finite, in order to obtain information about the structure of G.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
S.N. Černikov, Infinite groups with prescribed properties of their systems of
infinite subgroups, Dokl. Akad. Nauk SSSR, 159 (1964) 759-760 (Russian), Soviet Math. Dokl., 5 (1964), 1610-1611.
S.N. Černikov, Infinite nonabelian groups with minimal condition for
non-normal subgroups, Mat. Zametki, 6 (1969) 11-18 (Russian), Math. Notes, 6 (1969), 465-468.
S.N. Černikov, Infinite nonabelian groups with the minimal condition for
noninvariant abelian subgroups, Dokl. Akad. Nauk SSSR, 184 (1969) 786-789 (Russian), Soviet Math. Dokl., 10 (1969), 172-175.
subgroups, Riv. Mat. Pura Appl., 9 (1991), 49-59.
R.E. Phillips, J.S. Wilson, On certain minimal conditions for infinite groups,
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
S.N. Černikov, Infinite groups with prescribed properties of their systems of
infinite subgroups, Dokl. Akad. Nauk SSSR, 159 (1964) 759-760 (Russian), Soviet Math. Dokl., 5 (1964), 1610-1611.
S.N. Černikov, Infinite nonabelian groups with minimal condition for
non-normal subgroups, Mat. Zametki, 6 (1969) 11-18 (Russian), Math. Notes, 6 (1969), 465-468.
S.N. Černikov, Infinite nonabelian groups with the minimal condition for
noninvariant abelian subgroups, Dokl. Akad. Nauk SSSR, 184 (1969) 786-789 (Russian), Soviet Math. Dokl., 10 (1969), 172-175.
subgroups, Riv. Mat. Pura Appl., 9 (1991), 49-59.
R.E. Phillips, J.S. Wilson, On certain minimal conditions for infinite groups,
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Dipartimento di Matematica Università di Salerno via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy E-mail address : plongobardi@unisa.it
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
S.N. Černikov, Infinite groups with prescribed properties of their
systems of infinite subgroups, Dokl. Akad. Nauk SSSR, 159 (1964) 759-760 (Russian), Soviet Math. Dokl., 5 (1964), 1610-1611.
S.N. Černikov, Groups with given properties of systems of infinite
subgroups, Ukrain. Mat. Ž, 19 (1967) 111-131 (Russian), Ukrainian
S.N. Černikov, Infinite nonabelian groups with minimal condition
for non-normal subgroups, Mat. Zametki, 6 (1969) 11-18 (Russian),
S.N. Černikov, Infinite nonabelian groups with the minimal
condition for noninvariant abelian subgroups, Dokl. Akad. Nauk SSSR, 184 (1969) 786-789 (Russian), Soviet Math. Dokl., 10 (1969), 172-175.
V.S. Charin, On soluble groups of type A4, Mat. Sbornik 52
(1960), no. 3, 895-914.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
nonnormal subgroups, Riv. Mat. Pura Appl., 9 (1991), 49-59.
M.R. Dixon, L.A. Kurdachenko, Groups with the maximum
condition on non-nilpotent subgroups, J. Group Theory 4 (2001), 75-87.
M.R. Dixon, L.A. Kurdachenko, Locally nilpotent groups with
the maximum condition on non-nilpotent subgroups, Glasgow Math.
M.R. Dixon, L.A. Kurdachenko, Groups with the maximal
condition on non-BFC subgroups, Algebra Colloq. 10 (2003), 177-193.
M.R. Dixon, L.A. Kurdachenko, Groups with the maximal
condition on non-BFC subgroups II, Proc. Edinburgh Math. Soc. 45
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
M.R. Dixon, L.A. Kurdachenko, Groups with the maximal
condition on non FC-subgroups, Illinois J. Math. 47 no. 1/2 (2003), 157-172.
derived subgroups, J. London Math. Soc. 71 (2005), no. 2, 658-668.
groups, Ischia Group Theory 2004, Contemp. Math. Amer. Math. Soc., Providence, RI 402 (2006), 181-192.
L.A. Kurdachenko, P. Longobardi, M.Maj, Groups with finitely
many types of non-isomorphic non-abelian subgroups, in preparation.
L.A. Kurdachenko, H. Smith, Groups with the maximal
condition on non-subnormal subgroups, Boll. Unione Mat. Ital., Ser. B 10 (1996), 441-460.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
L.A. Kurdachenko, D.I. Zaicev, Groups with the maximum
condition for non-abelian subgroups, Ukrain. Mat. Zh. 43 (1991), 925-930 (Russian), English transl., Ukrainian Math. J. 43 (1991), 863-868.
two isomorphism classes of derived subgroups, Glasgow Math. J. 55 (2013), no. 3, 655-668.
few isomorphism classes of derived subgroups, Proc. of "Group Theory, Combinatorics, and Computing", Boca Raton-Florida,
finitely many isomorphism classes of derived subgroups, J. Algebra, 393 (2013), 102-119.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
A.I. Mal’cev, On certain classes of infinite soluble groups, Mat. Sb.
28 (1951), 567-588 (Russian), Amer. Math. Soc. Transl. (2) 2 (1956), 1-21.
Mathematics and its Applications, vol.70 Kluwer Academic Publishers, Dordrecht, 1989.
R.E. Phillips, J.S. Wilson, On certain minimal conditions for
infinite groups, J. Algebra 51 (1951), 41-68.
O.J. Schmidt, Infinite soluble groups, Mat. Sb., 17(59) (1945),
145-162 (Russian).
non-abelian subgroups, Rend. Math. Univ. Padova 97 (1997), 7-16.
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...
Patrizia Longobardi - University of Salerno Infinite groups with a finiteness conditions on non-abelian ...