Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee Improving Cayley’s Theorem for Groups of Outline Order p 4 Cayley’s Theorem Group Actions Sean McAfee P-Groups and an Algorithm for Finding ℓ University of Illinois at Chicago Results Undergraduate Mathematics Symposium – October 1, 2011
Improving Cayley’s Theorem for Groups of Order p 4 1 Cayley’s Theorem Sean McAfee Outline Cayley’s 2 Group Actions Theorem Group Actions P-Groups and an Algorithm 3 P-Groups and an Algorithm for Finding ℓ for Finding ℓ Results 4 Results
Statement of the Theorem Improving Cayley’s Theorem for Groups of Order p 4 Cayley’s Theorem: Let G be a group of order Sean McAfee n. Then G is isomorphic to a subgroup of S n . Outline Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
Statement of the Theorem Improving Cayley’s Theorem for Groups of Order p 4 Cayley’s Theorem: Let G be a group of order Sean McAfee n. Then G is isomorphic to a subgroup of S n . Outline Cayley’s Theorem This theorem tells us that every group can be understood as a Group Actions permutation group. However, it does little to tell us about the P-Groups and an Algorithm structure of G. for Finding ℓ Results
Statement of the Theorem Improving Cayley’s Theorem for Groups of Order p 4 Cayley’s Theorem: Let G be a group of order Sean McAfee n. Then G is isomorphic to a subgroup of S n . Outline Cayley’s Theorem This theorem tells us that every group can be understood as a Group Actions permutation group. However, it does little to tell us about the P-Groups and an Algorithm structure of G. for Finding ℓ Results Based on this information alone, all we can say is that G is contained in a group of size n!.
Can we do better? Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee A natural question to ask is, can this bound be improved? Outline Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
Can we do better? Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee A natural question to ask is, can this bound be improved? Outline Can we find integers m < n such that G is contained in S m ? Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
Can we do better? Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee A natural question to ask is, can this bound be improved? Outline Can we find integers m < n such that G is contained in S m ? Cayley’s Theorem Group Actions Can we find the least integer ℓ such that G is contained in S ℓ ? P-Groups and an Algorithm for Finding ℓ Results
Can we do better? Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee A natural question to ask is, can this bound be improved? Outline Can we find integers m < n such that G is contained in S m ? Cayley’s Theorem Group Actions Can we find the least integer ℓ such that G is contained in S ℓ ? P-Groups and an Algorithm for Finding ℓ In many cases we can, and we will see that this number is Results closely tied to the minimal normal subgroups of G.
Restating the Question Improving Cayley’s Theorem for It will be helpful to frame this problem in terms of group Groups of Order p 4 actions. Sean McAfee Outline Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
Restating the Question Improving Cayley’s Theorem for It will be helpful to frame this problem in terms of group Groups of Order p 4 actions. Sean McAfee Suppose G is isomorphic to a subgroup of S ℓ . This is the same Outline Cayley’s as saying that G acts faithfully on some set A of size ℓ : Theorem Group Actions G � A . P-Groups and an Algorithm for Finding ℓ Results
Restating the Question Improving Cayley’s Theorem for It will be helpful to frame this problem in terms of group Groups of Order p 4 actions. Sean McAfee Suppose G is isomorphic to a subgroup of S ℓ . This is the same Outline Cayley’s as saying that G acts faithfully on some set A of size ℓ : Theorem Group Actions G � A . P-Groups and an Algorithm for Finding ℓ This is equivalent to G acting on the disjoint union of orbits of Results A: G � O x 1 ⊔ . . . ⊔ O x k .
Restating the Question Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee For a given orbit O x i , we have a bijection between O x i and the Outline G collection of cosets Stab xi , where Stab x i = { g ∈ G | gx i = x i } . Cayley’s Theorem Thus our group action is equivalent to Group Actions P-Groups and an Algorithm for Finding ℓ Results
Restating the Question Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee For a given orbit O x i , we have a bijection between O x i and the Outline G collection of cosets Stab xi , where Stab x i = { g ∈ G | gx i = x i } . Cayley’s Theorem Thus our group action is equivalent to Group Actions P-Groups and G G G � Stab x 1 ⊔ . . . ⊔ Stab xk . an Algorithm for Finding ℓ Results
Restating the Question Improving Cayley’s Theorem for Groups of Order p 4 Now, since the stabilizer of an element of A is a subgroup of G, Sean McAfee we have that any group action can be represented by Outline Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
Restating the Question Improving Cayley’s Theorem for Groups of Order p 4 Now, since the stabilizer of an element of A is a subgroup of G, Sean McAfee we have that any group action can be represented by Outline G � G G Cayley’s H 1 ⊔ . . . ⊔ H k , Theorem Group Actions for some subgroups H 1 ,..., H k in G. P-Groups and an Algorithm for Finding ℓ Results
Restating the Question Improving Cayley’s Theorem for Groups of Order p 4 Now, since the stabilizer of an element of A is a subgroup of G, Sean McAfee we have that any group action can be represented by Outline G � G G Cayley’s H 1 ⊔ . . . ⊔ H k , Theorem Group Actions for some subgroups H 1 ,..., H k in G. P-Groups and an Algorithm for Finding ℓ It can be shown that this action is faithful if and only if the Results intersection of the H i ’s contains no nontrivial minimal normal subgroups of G.
Restating the Question Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee This means that in order to find a smallest ℓ with G isomorphic to a subgroup of S ℓ , we need a collection of subgroups { H i } of Outline G such that: Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
Restating the Question Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee This means that in order to find a smallest ℓ with G isomorphic to a subgroup of S ℓ , we need a collection of subgroups { H i } of Outline G such that: Cayley’s Theorem Group Actions 1 The intersection of the H i ’s does not contain a minimal P-Groups and normal subgroup of G. an Algorithm for Finding ℓ Results
Restating the Question Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee This means that in order to find a smallest ℓ with G isomorphic to a subgroup of S ℓ , we need a collection of subgroups { H i } of Outline G such that: Cayley’s Theorem Group Actions 1 The intersection of the H i ’s does not contain a minimal P-Groups and normal subgroup of G. an Algorithm for Finding ℓ � � � � � G � G Results � +...+ � is as small as possible. 2 � � � � H 1 H k
What Next? Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee How do we find such a collection of subgroups of G? Outline Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
What Next? Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee How do we find such a collection of subgroups of G? Outline This can be difficult in general. For an arbitrary group G, we Cayley’s Theorem can’t say much about what the minimal normal subgroups look Group Actions like or where they might be found inside of G. P-Groups and an Algorithm for Finding ℓ Results
What Next? Improving Cayley’s Theorem for Groups of Order p 4 Sean McAfee How do we find such a collection of subgroups of G? Outline This can be difficult in general. For an arbitrary group G, we Cayley’s Theorem can’t say much about what the minimal normal subgroups look Group Actions like or where they might be found inside of G. P-Groups and an Algorithm for Finding ℓ With groups of prime power order, however, our search is much Results easier..
Working with p-groups Improving Cayley’s Theorem for A group G of order p k has three properties which will be useful Groups of Order p 4 to us: Sean McAfee Outline Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
Working with p-groups Improving Cayley’s Theorem for A group G of order p k has three properties which will be useful Groups of Order p 4 to us: Sean McAfee 1 The minimal normal subgroups of G are all of order p. Outline Cayley’s Theorem Group Actions P-Groups and an Algorithm for Finding ℓ Results
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