s ssts t rs - - PowerPoint PPT Presentation
s ssts t rs t r rst r rs t rs
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❯♥✐✈❡rs✐t② ♦❢ ❆❜❡r❞❡❡♥
❇✐r♠✐♥❣❤❛♠✱ ✼t❤ ❆✉❣✉st ✷✵✶✼
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶ ✴ ✶✺
▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❚✇♦ s✉❜❣r♦✉♣s P, Q ≤ G ❛r❡ ❢✉s❡❞ ✐♥ G ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❡❧❡♠❡♥t g ∈ G s✉❝❤ t❤❛t P g = Q✳
❉❡✜♥✐t✐♦♥
▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ ❙②❧♦✇ ✲s✉❜❣r♦✉♣ ♦❢ ✳ ❚❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② ♦❢ ♦♥ ✐s t❤❡ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠ s❡ts ❛r❡✿ ❢♦r ❡✈❡r② ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✷ ✴ ✶✺
▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✳ ❚✇♦ s✉❜❣r♦✉♣s P, Q ≤ G ❛r❡ ❢✉s❡❞ ✐♥ G ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❡❧❡♠❡♥t g ∈ G s✉❝❤ t❤❛t P g = Q✳
❉❡✜♥✐t✐♦♥
▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t S ❜❡ ❛ ❙②❧♦✇ p✲s✉❜❣r♦✉♣ ♦❢ G✳ ❚❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② ♦❢ G ♦♥ S ✐s t❤❡ ❝❛t❡❣♦r② FS(G) ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ S ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠ s❡ts ❛r❡✿ MorFS(G)(P, Q) = HomG(P, Q) = {cg|P : P → Q|g ∈ G, P g ≤ Q}, ❢♦r ❡✈❡r② P, Q ≤ S✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✷ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ s✉❝❤ t❤❛t ❙②❧ ✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜②
❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ✭ ❡ss❡♥t✐❛❧✮ ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧p(G)✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜②
G = D8
❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ✭ ❡ss❡♥t✐❛❧✮ ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧p(G)✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts g ∈ G t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ S✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② Inn(D8) : FD8(D8) = Inn(D8)
G = D8
❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ✭ ❡ss❡♥t✐❛❧✮ ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧p(G)✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜②
G = Sym(4) = S4
a = (1234), x = (13).
S4
b ❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ✭ ❡ss❡♥t✐❛❧✮ ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧p(G)✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜②
G = Sym(4) = S4
a = (1234), x = (13). b = (123) (a2)b = ((13)(24))(123) = (12)(34) = ax
S4
b ❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② Inn(D8) ❛♥❞ Aut(E) ∼ = SL2(2) ∼ = S3 FD8(S4) = Inn(D8), Aut(E) ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ✭ ❡ss❡♥t✐❛❧✮ ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧p(G)✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜②
S4
b ❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞
G = A6
a = (1234)(56), x = (13)(56)✳ c = (25)(46)
A6
c
(a2)c = ((13)(24))(25)(46) = (13)(56) = x ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ✭ ❡ss❡♥t✐❛❧✮ ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧p(G)✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜②
S4
b ❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞
G = A6
a = (1234)(56), x = (13)(56)✳ c = (25)(46)
A6
c
(a2)c = ((13)(24))(25)(46) = (13)(56) = x ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② Inn(D8)✱ Aut(E) ❛♥❞ Aut(P) FD8(A6) = Inn(D8), Aut(E), Aut(P)
E P
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ ❄ ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ ✭ ❡ss❡♥t✐❛❧✮ ✭ ❛♥❞ ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ s✉❝❤ t❤❛t ❙②❧ ✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜②
❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞
E P
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ D8❄ −FD8(D8) ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ −FD8(S4) ✭E ❡ss❡♥t✐❛❧✮ −FD8(A6) ✭E ❛♥❞ P ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
P✐❝❦ ❛ p✲❣r♦✉♣ S✳ P✐❝❦ ❛ ✜♥✐t❡ ❣r♦✉♣ s✉❝❤ t❤❛t ❙②❧ ✳ D8
a3x ax a2 x a2x a a3 1
S = D8
D8 := a, x | a4 = x2 = 1, ax = a3 ❈♦♥s✐❞❡r t❤❡ ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s ❜② ❡❧❡♠❡♥ts t❤❛t ❢✉s❡ s♦♠❡ ❡❧❡♠❡♥ts✴s✉❜❣r♦✉♣s ♦❢ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜②
❊
❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛♥❞ ✳ ❢✉s✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ✱ ❛♥❞
E P
❍♦✇ ♠❛♥② ✇❛②s t♦ ❢✉s❡ ❡❧❡♠❡♥ts ♦❢ D8❄ −FD8(D8) ✭♥♦ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣s✮ −FD8(S4) ✭E ❡ss❡♥t✐❛❧✮ −FD8(A6) ✭E ❛♥❞ P ❡ss❡♥t✐❛❧✮
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✸ ✴ ✶✺
▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t S ❜❡ ❛ p✲❣r♦✉♣✳ ❈❤♦♦s❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r♦✉♣s ♦❢ S t❤❛t ✧❜❡❤❛✈❡✧ ❛s ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s✳
❉❡✜♥✐t✐♦♥ ✭❋✉s✐♦♥ ❙②st❡♠✮
❆ ❋✉s✐♦♥ s②st❡♠ F ♦♥ S ✐s ❛ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ S ❛♥❞ ✇✐t❤ ♠♦r♣❤✐s♠ s❡ts HomF(P, Q) ⊆ Inj(P, Q) s✉❝❤ t❤❛t
✶ HomS(P, Q) ⊆ HomF(P, Q)✱ ✷ ❡❛❝❤ ϕ ∈ HomF(P, Q) ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐s♦♠♦r♣❤✐s♠
α ∈ Mor(F) ❛♥❞ ❛♥ ✐♥❝❧✉s✐♦♥ β ∈ Mor(F)✳
❆ ❋✉s✐♦♥ s②st❡♠ ✐s ❙❛t✉r❛t❡❞ ✐❢ ✐t s❛t✐s✜❡s ❝❡rt❛✐♥ ❡①tr❛ ♣r♦♣❡rt✐❡s ✭❙②❧♦✇ ❛♥❞ ❡①t❡♥s✐♦♥ ♣r♦♣❡rt✐❡s✮✳ ■❢ ❙②❧ t❤❡♥ t❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ ✳ ■❢ ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ ❛♥❞ t❤❡r❡ ✐s ♥♦ ✜♥✐t❡ ❣r♦✉♣ s✉❝❤ t❤❛t ❙②❧ ❛♥❞ ✱ t❤❡♥ ✐s ❝❛❧❧❡❞ ❡①♦t✐❝✳
◗✉❡st✐♦♥ ✭s✉❣❣❡st❡❞ ❜② ❖❧✐✈❡r✮
❚r② t♦ ❜❡tt❡r ✉♥❞❡rst❛♥❞ ❤♦✇ ❡①♦t✐❝ ❢✉s✐♦♥ s②st❡♠s ❛r✐s❡ ❛t ♦❞❞ ♣r✐♠❡s✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✹ ✴ ✶✺
▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t S ❜❡ ❛ p✲❣r♦✉♣✳ ❈❤♦♦s❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r♦✉♣s ♦❢ S t❤❛t ✧❜❡❤❛✈❡✧ ❛s ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s✳
❉❡✜♥✐t✐♦♥ ✭❋✉s✐♦♥ ❙②st❡♠✮
❆ ❋✉s✐♦♥ s②st❡♠ F ♦♥ S ✐s ❛ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ S ❛♥❞ ✇✐t❤ ♠♦r♣❤✐s♠ s❡ts HomF(P, Q) ⊆ Inj(P, Q) s✉❝❤ t❤❛t
✶ HomS(P, Q) ⊆ HomF(P, Q)✱ ✷ ❡❛❝❤ ϕ ∈ HomF(P, Q) ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐s♦♠♦r♣❤✐s♠
α ∈ Mor(F) ❛♥❞ ❛♥ ✐♥❝❧✉s✐♦♥ β ∈ Mor(F)✳
❆ ❋✉s✐♦♥ s②st❡♠ ✐s ❙❛t✉r❛t❡❞ ✐❢ ✐t s❛t✐s✜❡s ❝❡rt❛✐♥ ❡①tr❛ ♣r♦♣❡rt✐❡s ✭❙②❧♦✇ ❛♥❞ ❡①t❡♥s✐♦♥ ♣r♦♣❡rt✐❡s✮✳ ■❢ ❙②❧ t❤❡♥ t❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ ✳ ■❢ ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ ❛♥❞ t❤❡r❡ ✐s ♥♦ ✜♥✐t❡ ❣r♦✉♣ s✉❝❤ t❤❛t ❙②❧ ❛♥❞ ✱ t❤❡♥ ✐s ❝❛❧❧❡❞ ❡①♦t✐❝✳
◗✉❡st✐♦♥ ✭s✉❣❣❡st❡❞ ❜② ❖❧✐✈❡r✮
❚r② t♦ ❜❡tt❡r ✉♥❞❡rst❛♥❞ ❤♦✇ ❡①♦t✐❝ ❢✉s✐♦♥ s②st❡♠s ❛r✐s❡ ❛t ♦❞❞ ♣r✐♠❡s✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✹ ✴ ✶✺
▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t S ❜❡ ❛ p✲❣r♦✉♣✳ ❈❤♦♦s❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r♦✉♣s ♦❢ S t❤❛t ✧❜❡❤❛✈❡✧ ❛s ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s✳
❉❡✜♥✐t✐♦♥ ✭❋✉s✐♦♥ ❙②st❡♠✮
❆ ❋✉s✐♦♥ s②st❡♠ F ♦♥ S ✐s ❛ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ S ❛♥❞ ✇✐t❤ ♠♦r♣❤✐s♠ s❡ts HomF(P, Q) ⊆ Inj(P, Q) s✉❝❤ t❤❛t
✶ HomS(P, Q) ⊆ HomF(P, Q)✱ ✷ ❡❛❝❤ ϕ ∈ HomF(P, Q) ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐s♦♠♦r♣❤✐s♠
α ∈ Mor(F) ❛♥❞ ❛♥ ✐♥❝❧✉s✐♦♥ β ∈ Mor(F)✳
❆ ❋✉s✐♦♥ s②st❡♠ ✐s ❙❛t✉r❛t❡❞ ✐❢ ✐t s❛t✐s✜❡s ❝❡rt❛✐♥ ❡①tr❛ ♣r♦♣❡rt✐❡s ✭❙②❧♦✇ ❛♥❞ ❡①t❡♥s✐♦♥ ♣r♦♣❡rt✐❡s✮✳ ■❢ S ∈ ❙②❧p(G) t❤❡♥ t❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② FS(G) ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ S✳ ■❢ ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ ❛♥❞ t❤❡r❡ ✐s ♥♦ ✜♥✐t❡ ❣r♦✉♣ s✉❝❤ t❤❛t ❙②❧ ❛♥❞ ✱ t❤❡♥ ✐s ❝❛❧❧❡❞ ❡①♦t✐❝✳
◗✉❡st✐♦♥ ✭s✉❣❣❡st❡❞ ❜② ❖❧✐✈❡r✮
❚r② t♦ ❜❡tt❡r ✉♥❞❡rst❛♥❞ ❤♦✇ ❡①♦t✐❝ ❢✉s✐♦♥ s②st❡♠s ❛r✐s❡ ❛t ♦❞❞ ♣r✐♠❡s✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✹ ✴ ✶✺
▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t S ❜❡ ❛ p✲❣r♦✉♣✳ ❈❤♦♦s❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r♦✉♣s ♦❢ S t❤❛t ✧❜❡❤❛✈❡✧ ❛s ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s✳
❉❡✜♥✐t✐♦♥ ✭❋✉s✐♦♥ ❙②st❡♠✮
❆ ❋✉s✐♦♥ s②st❡♠ F ♦♥ S ✐s ❛ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ S ❛♥❞ ✇✐t❤ ♠♦r♣❤✐s♠ s❡ts HomF(P, Q) ⊆ Inj(P, Q) s✉❝❤ t❤❛t
✶ HomS(P, Q) ⊆ HomF(P, Q)✱ ✷ ❡❛❝❤ ϕ ∈ HomF(P, Q) ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐s♦♠♦r♣❤✐s♠
α ∈ Mor(F) ❛♥❞ ❛♥ ✐♥❝❧✉s✐♦♥ β ∈ Mor(F)✳
❆ ❋✉s✐♦♥ s②st❡♠ ✐s ❙❛t✉r❛t❡❞ ✐❢ ✐t s❛t✐s✜❡s ❝❡rt❛✐♥ ❡①tr❛ ♣r♦♣❡rt✐❡s ✭❙②❧♦✇ ❛♥❞ ❡①t❡♥s✐♦♥ ♣r♦♣❡rt✐❡s✮✳ ■❢ S ∈ ❙②❧p(G) t❤❡♥ t❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② FS(G) ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ S✳ ■❢ F ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ S ❛♥❞ t❤❡r❡ ✐s ♥♦ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧p(G) ❛♥❞ F = FS(G)✱ t❤❡♥ F ✐s ❝❛❧❧❡❞ ❡①♦t✐❝✳
◗✉❡st✐♦♥ ✭s✉❣❣❡st❡❞ ❜② ❖❧✐✈❡r✮
❚r② t♦ ❜❡tt❡r ✉♥❞❡rst❛♥❞ ❤♦✇ ❡①♦t✐❝ ❢✉s✐♦♥ s②st❡♠s ❛r✐s❡ ❛t ♦❞❞ ♣r✐♠❡s✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✹ ✴ ✶✺
▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t S ❜❡ ❛ p✲❣r♦✉♣✳ ❈❤♦♦s❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r♦✉♣s ♦❢ S t❤❛t ✧❜❡❤❛✈❡✧ ❛s ❝♦♥❥✉❣❛t✐♦♥ ♠❛♣s✳
❉❡✜♥✐t✐♦♥ ✭❋✉s✐♦♥ ❙②st❡♠✮
❆ ❋✉s✐♦♥ s②st❡♠ F ♦♥ S ✐s ❛ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ t❤❡ s✉❜❣r♦✉♣s ♦❢ S ❛♥❞ ✇✐t❤ ♠♦r♣❤✐s♠ s❡ts HomF(P, Q) ⊆ Inj(P, Q) s✉❝❤ t❤❛t
✶ HomS(P, Q) ⊆ HomF(P, Q)✱ ✷ ❡❛❝❤ ϕ ∈ HomF(P, Q) ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛♥ ✐s♦♠♦r♣❤✐s♠
α ∈ Mor(F) ❛♥❞ ❛♥ ✐♥❝❧✉s✐♦♥ β ∈ Mor(F)✳
❆ ❋✉s✐♦♥ s②st❡♠ ✐s ❙❛t✉r❛t❡❞ ✐❢ ✐t s❛t✐s✜❡s ❝❡rt❛✐♥ ❡①tr❛ ♣r♦♣❡rt✐❡s ✭❙②❧♦✇ ❛♥❞ ❡①t❡♥s✐♦♥ ♣r♦♣❡rt✐❡s✮✳ ■❢ S ∈ ❙②❧p(G) t❤❡♥ t❤❡ ❢✉s✐♦♥ ❝❛t❡❣♦r② FS(G) ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ S✳ ■❢ F ✐s ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ S ❛♥❞ t❤❡r❡ ✐s ♥♦ ✜♥✐t❡ ❣r♦✉♣ G s✉❝❤ t❤❛t S ∈ ❙②❧p(G) ❛♥❞ F = FS(G)✱ t❤❡♥ F ✐s ❝❛❧❧❡❞ ❡①♦t✐❝✳
◗✉❡st✐♦♥ ✭s✉❣❣❡st❡❞ ❜② ❖❧✐✈❡r✮
❚r② t♦ ❜❡tt❡r ✉♥❞❡rst❛♥❞ ❤♦✇ ❡①♦t✐❝ ❢✉s✐♦♥ s②st❡♠s ❛r✐s❡ ❛t ♦❞❞ ♣r✐♠❡s✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✹ ✴ ✶✺
◗✉❡st✐♦♥
♦♥ t❤❡♠❄ ❆♥ ✐♥t❡r❡st✐♥❣ ❝❧❛ss t♦ st✉❞② ✐s t❤❡ ❝❧❛ss ♦❢ ✲❣r♦✉♣s ❤❛✈✐♥❣ s♠❛❧❧ s❡❝t✐♦♥❛❧ r❛♥❦✳ ❆ ✲❣r♦✉♣ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ✐❢ ❡✈❡r② ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ s❡❝t✐♦♥ ♦❢ ❤❛s ♦r❞❡r ❛t ♠♦st ✭❡q✉✐✈❛❧❡♥t❧② ✐❢ ❡✈❡r② s✉❜❣r♦✉♣ ♦❢ ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② ❡❧❡♠❡♥ts✮✳ ❖❧✐✈❡r✱ ✷✵✶✻ ✭▼❡♠♦✐r ♦❢ t❤❡ ❆▼❙✮✿ ✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❛t ♠♦st ❀ ❉✐❛③✱ ❘✉✐③ ❛♥❞ ❱✐r✉❡❧✱ ✷✵✵✼✿ ♦❞❞✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❀
✱ s❡❝t✐♦♥❛❧ r❛♥❦ ✳
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ ✲❣r♦✉♣ ✳ ■❢ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ❛♥❞ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ s✉❝❤ t❤❛t ❡✐t❤❡r ♦r ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✺ ✴ ✶✺
◗✉❡st✐♦♥
♦♥ t❤❡♠❄ ❆♥ ✐♥t❡r❡st✐♥❣ ❝❧❛ss t♦ st✉❞② ✐s t❤❡ ❝❧❛ss ♦❢ p✲❣r♦✉♣s ❤❛✈✐♥❣ s♠❛❧❧ s❡❝t✐♦♥❛❧ r❛♥❦✳ ❆ ✲❣r♦✉♣ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ✐❢ ❡✈❡r② ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ s❡❝t✐♦♥ ♦❢ ❤❛s ♦r❞❡r ❛t ♠♦st ✭❡q✉✐✈❛❧❡♥t❧② ✐❢ ❡✈❡r② s✉❜❣r♦✉♣ ♦❢ ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② ❡❧❡♠❡♥ts✮✳ ❖❧✐✈❡r✱ ✷✵✶✻ ✭▼❡♠♦✐r ♦❢ t❤❡ ❆▼❙✮✿ ✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❛t ♠♦st ❀ ❉✐❛③✱ ❘✉✐③ ❛♥❞ ❱✐r✉❡❧✱ ✷✵✵✼✿ ♦❞❞✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❀
✱ s❡❝t✐♦♥❛❧ r❛♥❦ ✳
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ ✲❣r♦✉♣ ✳ ■❢ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ❛♥❞ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ s✉❝❤ t❤❛t ❡✐t❤❡r ♦r ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✺ ✴ ✶✺
◗✉❡st✐♦♥
♦♥ t❤❡♠❄ ❆♥ ✐♥t❡r❡st✐♥❣ ❝❧❛ss t♦ st✉❞② ✐s t❤❡ ❝❧❛ss ♦❢ p✲❣r♦✉♣s ❤❛✈✐♥❣ s♠❛❧❧ s❡❝t✐♦♥❛❧ r❛♥❦✳ ❆ p✲❣r♦✉♣ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ✐❢ ❡✈❡r② ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ s❡❝t✐♦♥ P/Q ♦❢ S ❤❛s ♦r❞❡r ❛t ♠♦st pk ✭❡q✉✐✈❛❧❡♥t❧② ✐❢ ❡✈❡r② s✉❜❣r♦✉♣ ♦❢ S ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② k ❡❧❡♠❡♥ts✮✳ ❖❧✐✈❡r✱ ✷✵✶✻ ✭▼❡♠♦✐r ♦❢ t❤❡ ❆▼❙✮✿ ✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❛t ♠♦st ❀ ❉✐❛③✱ ❘✉✐③ ❛♥❞ ❱✐r✉❡❧✱ ✷✵✵✼✿ ♦❞❞✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❀
✱ s❡❝t✐♦♥❛❧ r❛♥❦ ✳
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ ✲❣r♦✉♣ ✳ ■❢ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ❛♥❞ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ s✉❝❤ t❤❛t ❡✐t❤❡r ♦r ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✺ ✴ ✶✺
◗✉❡st✐♦♥
♦♥ t❤❡♠❄ ❆♥ ✐♥t❡r❡st✐♥❣ ❝❧❛ss t♦ st✉❞② ✐s t❤❡ ❝❧❛ss ♦❢ p✲❣r♦✉♣s ❤❛✈✐♥❣ s♠❛❧❧ s❡❝t✐♦♥❛❧ r❛♥❦✳ ❆ p✲❣r♦✉♣ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ✐❢ ❡✈❡r② ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ s❡❝t✐♦♥ P/Q ♦❢ S ❤❛s ♦r❞❡r ❛t ♠♦st pk ✭❡q✉✐✈❛❧❡♥t❧② ✐❢ ❡✈❡r② s✉❜❣r♦✉♣ ♦❢ S ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② k ❡❧❡♠❡♥ts✮✳ ❖❧✐✈❡r✱ ✷✵✶✻ ✭▼❡♠♦✐r ♦❢ t❤❡ ❆▼❙✮✿ p = 2✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❛t ♠♦st 4❀ ❉✐❛③✱ ❘✉✐③ ❛♥❞ ❱✐r✉❡❧✱ ✷✵✵✼✿ ♦❞❞✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❀
✱ s❡❝t✐♦♥❛❧ r❛♥❦ ✳
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ ✲❣r♦✉♣ ✳ ■❢ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ❛♥❞ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ s✉❝❤ t❤❛t ❡✐t❤❡r ♦r ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✺ ✴ ✶✺
◗✉❡st✐♦♥
♦♥ t❤❡♠❄ ❆♥ ✐♥t❡r❡st✐♥❣ ❝❧❛ss t♦ st✉❞② ✐s t❤❡ ❝❧❛ss ♦❢ p✲❣r♦✉♣s ❤❛✈✐♥❣ s♠❛❧❧ s❡❝t✐♦♥❛❧ r❛♥❦✳ ❆ p✲❣r♦✉♣ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ✐❢ ❡✈❡r② ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ s❡❝t✐♦♥ P/Q ♦❢ S ❤❛s ♦r❞❡r ❛t ♠♦st pk ✭❡q✉✐✈❛❧❡♥t❧② ✐❢ ❡✈❡r② s✉❜❣r♦✉♣ ♦❢ S ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② k ❡❧❡♠❡♥ts✮✳ ❖❧✐✈❡r✱ ✷✵✶✻ ✭▼❡♠♦✐r ♦❢ t❤❡ ❆▼❙✮✿ p = 2✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❛t ♠♦st 4❀ ❉✐❛③✱ ❘✉✐③ ❛♥❞ ❱✐r✉❡❧✱ ✷✵✵✼✿ p ♦❞❞✱ s❡❝t✐♦♥❛❧ r❛♥❦ 2❀
✱ s❡❝t✐♦♥❛❧ r❛♥❦ ✳
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ ✲❣r♦✉♣ ✳ ■❢ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ❛♥❞ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ s✉❝❤ t❤❛t ❡✐t❤❡r ♦r ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✺ ✴ ✶✺
◗✉❡st✐♦♥
♦♥ t❤❡♠❄ ❆♥ ✐♥t❡r❡st✐♥❣ ❝❧❛ss t♦ st✉❞② ✐s t❤❡ ❝❧❛ss ♦❢ p✲❣r♦✉♣s ❤❛✈✐♥❣ s♠❛❧❧ s❡❝t✐♦♥❛❧ r❛♥❦✳ ❆ p✲❣r♦✉♣ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ✐❢ ❡✈❡r② ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ s❡❝t✐♦♥ P/Q ♦❢ S ❤❛s ♦r❞❡r ❛t ♠♦st pk ✭❡q✉✐✈❛❧❡♥t❧② ✐❢ ❡✈❡r② s✉❜❣r♦✉♣ ♦❢ S ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② k ❡❧❡♠❡♥ts✮✳ ❖❧✐✈❡r✱ ✷✵✶✻ ✭▼❡♠♦✐r ♦❢ t❤❡ ❆▼❙✮✿ p = 2✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❛t ♠♦st 4❀ ❉✐❛③✱ ❘✉✐③ ❛♥❞ ❱✐r✉❡❧✱ ✷✵✵✼✿ p ♦❞❞✱ s❡❝t✐♦♥❛❧ r❛♥❦ 2❀
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ ✲❣r♦✉♣ ✳ ■❢ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ❛♥❞ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ s✉❝❤ t❤❛t ❡✐t❤❡r ♦r ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✺ ✴ ✶✺
◗✉❡st✐♦♥
♦♥ t❤❡♠❄ ❆♥ ✐♥t❡r❡st✐♥❣ ❝❧❛ss t♦ st✉❞② ✐s t❤❡ ❝❧❛ss ♦❢ p✲❣r♦✉♣s ❤❛✈✐♥❣ s♠❛❧❧ s❡❝t✐♦♥❛❧ r❛♥❦✳ ❆ p✲❣r♦✉♣ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ✐❢ ❡✈❡r② ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ s❡❝t✐♦♥ P/Q ♦❢ S ❤❛s ♦r❞❡r ❛t ♠♦st pk ✭❡q✉✐✈❛❧❡♥t❧② ✐❢ ❡✈❡r② s✉❜❣r♦✉♣ ♦❢ S ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② k ❡❧❡♠❡♥ts✮✳ ❖❧✐✈❡r✱ ✷✵✶✻ ✭▼❡♠♦✐r ♦❢ t❤❡ ❆▼❙✮✿ p = 2✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❛t ♠♦st 4❀ ❉✐❛③✱ ❘✉✐③ ❛♥❞ ❱✐r✉❡❧✱ ✷✵✵✼✿ p ♦❞❞✱ s❡❝t✐♦♥❛❧ r❛♥❦ 2❀
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ ✲❣r♦✉♣ ✳ ■❢ ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ❛♥❞ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ s✉❝❤ t❤❛t ❡✐t❤❡r ♦r ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✺ ✴ ✶✺
◗✉❡st✐♦♥
♦♥ t❤❡♠❄ ❆♥ ✐♥t❡r❡st✐♥❣ ❝❧❛ss t♦ st✉❞② ✐s t❤❡ ❝❧❛ss ♦❢ p✲❣r♦✉♣s ❤❛✈✐♥❣ s♠❛❧❧ s❡❝t✐♦♥❛❧ r❛♥❦✳ ❆ p✲❣r♦✉♣ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ✐❢ ❡✈❡r② ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ s❡❝t✐♦♥ P/Q ♦❢ S ❤❛s ♦r❞❡r ❛t ♠♦st pk ✭❡q✉✐✈❛❧❡♥t❧② ✐❢ ❡✈❡r② s✉❜❣r♦✉♣ ♦❢ S ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② k ❡❧❡♠❡♥ts✮✳ ❖❧✐✈❡r✱ ✷✵✶✻ ✭▼❡♠♦✐r ♦❢ t❤❡ ❆▼❙✮✿ p = 2✱ s❡❝t✐♦♥❛❧ r❛♥❦ ❛t ♠♦st 4❀ ❉✐❛③✱ ❘✉✐③ ❛♥❞ ❱✐r✉❡❧✱ ✷✵✵✼✿ p ♦❞❞✱ s❡❝t✐♦♥❛❧ r❛♥❦ 2❀
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t p ≥ 5 ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ■❢ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ 3 ❛♥❞ Op(F) = 1 t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ E ♦❢ S s✉❝❤ t❤❛t ❡✐t❤❡r E ∼ = Cp × Cp ♦r E ∼ = p1+2
+
✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✺ ✴ ✶✺
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t p ≥ 5 ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ■❢ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ 3 ❛♥❞ Op(F) = 1 t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ E ♦❢ S s✉❝❤ t❤❛t ❡✐t❤❡r E ∼ = Cp × Cp ♦r E ∼ = p1+2
+
✳ ▲❡t ✳ ❛♥❞ ❛r❡ t❤❡ ♦♥❧② s✉❜❣r♦✉♣s ♦❢ t❤❛t ❝❛♥ ❜❡ ❡ss❡♥t✐❛❧ ❛♥❞ ✳ ❈❛♥ ✇❡ ❝❤❛r❛❝t❡r✐③❡ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ❛♥ ❡ss❡♥t✐❛❧ s✉❜✲ ❣r♦✉♣ t❤❛t ✐s ❡✐t❤❡r ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r ♦r ♥♦♥✲❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r ❛♥❞ ❡①♣♦♥❡♥t ❄
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✻ ✴ ✶✺
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t p ≥ 5 ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ■❢ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ 3 ❛♥❞ Op(F) = 1 t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ E ♦❢ S s✉❝❤ t❤❛t ❡✐t❤❡r E ∼ = Cp × Cp ♦r E ∼ = p1+2
+
✳ ▲❡t S = D8✳ E ❛♥❞ P ❛r❡ t❤❡ ♦♥❧② s✉❜❣r♦✉♣s ♦❢ S t❤❛t ❝❛♥ ❜❡ ❡ss❡♥t✐❛❧ ❛♥❞ E ∼ = P ∼ = C2 × C2✳ D8
a3x ax a2 x a2x a a3 1 E P
❈❛♥ ✇❡ ❝❤❛r❛❝t❡r✐③❡ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ❛♥ ❡ss❡♥t✐❛❧ s✉❜✲ ❣r♦✉♣ t❤❛t ✐s ❡✐t❤❡r ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r ♦r ♥♦♥✲❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r ❛♥❞ ❡①♣♦♥❡♥t ❄
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✻ ✴ ✶✺
❚❤❡♦r❡♠ ✶ ✭●✳✱ ✷✵✶✻✮
▲❡t p ≥ 5 ❜❡ ❛ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ■❢ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ 3 ❛♥❞ Op(F) = 1 t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ E ♦❢ S s✉❝❤ t❤❛t ❡✐t❤❡r E ∼ = Cp × Cp ♦r E ∼ = p1+2
+
✳ ▲❡t S = D8✳ E ❛♥❞ P ❛r❡ t❤❡ ♦♥❧② s✉❜❣r♦✉♣s ♦❢ S t❤❛t ❝❛♥ ❜❡ ❡ss❡♥t✐❛❧ ❛♥❞ E ∼ = P ∼ = C2 × C2✳ D8
a3x ax a2 x a2x a a3 1 E P
❈❛♥ ✇❡ ❝❤❛r❛❝t❡r✐③❡ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ❛♥ ❡ss❡♥t✐❛❧ s✉❜✲ ❣r♦✉♣ t❤❛t ✐s ❡✐t❤❡r ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p2 ♦r ♥♦♥✲❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p3 ❛♥❞ ❡①♣♦♥❡♥t p❄
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✻ ✴ ✶✺
▲❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ❆ ♣❡❛r❧ ✐s ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ E ♦❢ S t❤❛t ✐s ❡✐t❤❡r ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p2 ✭E ∼ = Cp × Cp) ♦r ♥♦♥✲❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p3 ❛♥❞ ❡①♣♦♥❡♥t p ✭✐❢ p ✐s ♦❞❞ t❤❡♥ E ∼ = p1+2
+
✮✳ Pr♦♣❡rt② ✶✿ ❊ss❡♥t✐❛❧ s✉❜❣r♦✉♣s ♦❢ ❛r❡ s❡❧❢✲❝❡♥tr❛❧✐③✐♥❣ ✐♥ ✳ ❚❤❡♦r❡♠ ✭❙✉③✉❦✐✮✿ ■❢ ❛ ✲❣r♦✉♣ ❝♦♥t❛✐♥s ❛ s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r s✉❝❤ t❤❛t t❤❡♥ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❚❤❡♦r❡♠
■❢ ❝♦♥t❛✐♥s ❛ ♣❡❛r❧ t❤❡♥ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✼ ✴ ✶✺
▲❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ❆ ♣❡❛r❧ ✐s ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ E ♦❢ S t❤❛t ✐s ❡✐t❤❡r ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p2 ✭E ∼ = Cp × Cp) ♦r ♥♦♥✲❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p3 ❛♥❞ ❡①♣♦♥❡♥t p ✭✐❢ p ✐s ♦❞❞ t❤❡♥ E ∼ = p1+2
+
✮✳ Pr♦♣❡rt② ✶✿ ❊ss❡♥t✐❛❧ s✉❜❣r♦✉♣s ♦❢ S ❛r❡ s❡❧❢✲❝❡♥tr❛❧✐③✐♥❣ ✐♥ S✳ ❚❤❡♦r❡♠ ✭❙✉③✉❦✐✮✿ ■❢ ❛ ✲❣r♦✉♣ ❝♦♥t❛✐♥s ❛ s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r s✉❝❤ t❤❛t t❤❡♥ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❚❤❡♦r❡♠
■❢ ❝♦♥t❛✐♥s ❛ ♣❡❛r❧ t❤❡♥ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✼ ✴ ✶✺
▲❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ❆ ♣❡❛r❧ ✐s ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ E ♦❢ S t❤❛t ✐s ❡✐t❤❡r ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p2 ✭E ∼ = Cp × Cp) ♦r ♥♦♥✲❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p3 ❛♥❞ ❡①♣♦♥❡♥t p ✭✐❢ p ✐s ♦❞❞ t❤❡♥ E ∼ = p1+2
+
✮✳ Pr♦♣❡rt② ✶✿ ❊ss❡♥t✐❛❧ s✉❜❣r♦✉♣s ♦❢ S ❛r❡ s❡❧❢✲❝❡♥tr❛❧✐③✐♥❣ ✐♥ S✳ ❚❤❡♦r❡♠ ✭❙✉③✉❦✐✮✿ ■❢ ❛ p✲❣r♦✉♣ S ❝♦♥t❛✐♥s ❛ s✉❜❣r♦✉♣ E ♦❢ ♦r❞❡r p2 s✉❝❤ t❤❛t CS(E) = E t❤❡♥ S ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❚❤❡♦r❡♠
■❢ ❝♦♥t❛✐♥s ❛ ♣❡❛r❧ t❤❡♥ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✼ ✴ ✶✺
▲❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ❆ ♣❡❛r❧ ✐s ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ E ♦❢ S t❤❛t ✐s ❡✐t❤❡r ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p2 ✭E ∼ = Cp × Cp) ♦r ♥♦♥✲❛❜❡❧✐❛♥ ♦❢ ♦r❞❡r p3 ❛♥❞ ❡①♣♦♥❡♥t p ✭✐❢ p ✐s ♦❞❞ t❤❡♥ E ∼ = p1+2
+
✮✳ Pr♦♣❡rt② ✶✿ ❊ss❡♥t✐❛❧ s✉❜❣r♦✉♣s ♦❢ S ❛r❡ s❡❧❢✲❝❡♥tr❛❧✐③✐♥❣ ✐♥ S✳ ❚❤❡♦r❡♠ ✭❙✉③✉❦✐✮✿ ■❢ ❛ p✲❣r♦✉♣ S ❝♦♥t❛✐♥s ❛ s✉❜❣r♦✉♣ E ♦❢ ♦r❞❡r p2 s✉❝❤ t❤❛t CS(E) = E t❤❡♥ S ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❚❤❡♦r❡♠
■❢ F ❝♦♥t❛✐♥s ❛ ♣❡❛r❧ t❤❡♥ S ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✼ ✴ ✶✺
1 = Sn S = Zn−1(S)
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t ✱ ❢♦r ❡✈❡r② ❛♥❞ Pr♦♣❡rt✐❡s ♦❢ ✿ ❀ ❀ ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ ❀ ❛♥❞ ❢♦r ❡✈❡r② ✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ✐s t❤❡ ❣r♦✉♣ ✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ ✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn S = Zn−1(S) Sn−1 = Z(S) Sn−2 = Z2(S) S3 = Zn−3(S) S2 = Zn−2(S) S1 S4 = Zn−4(S)
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t S2 = [S, S]✱ Si = [Si−1, S] ❢♦r ❡✈❡r② i ≥ 3 ❛♥❞ S1 = CS(S2/S4). Pr♦♣❡rt✐❡s ♦❢ ✿ ❀ ❀ ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ ❀ ❛♥❞ ❢♦r ❡✈❡r② ✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ✐s t❤❡ ❣r♦✉♣ ✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ ✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn S = Zn−1(S) Sn−1 = Z(S) Sn−2 = Z2(S) S3 = Zn−3(S) S2 = Zn−2(S) S1 S4 = Zn−4(S)
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t S2 = [S, S]✱ Si = [Si−1, S] ❢♦r ❡✈❡r② i ≥ 3 ❛♥❞ S1 = CS(S2/S4). Pr♦♣❡rt✐❡s ♦❢ S1✿ S2 ≤ S1❀ ❀ ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ ❀ ❛♥❞ ❢♦r ❡✈❡r② ✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ✐s t❤❡ ❣r♦✉♣ ✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ ✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn S = Zn−1(S) Sn−1 = Z(S) Sn−2 = Z2(S) S3 = Zn−3(S) S2 = Zn−2(S) S1 S4 = Zn−4(S)
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t S2 = [S, S]✱ Si = [Si−1, S] ❢♦r ❡✈❡r② i ≥ 3 ❛♥❞ S1 = CS(S2/S4). Pr♦♣❡rt✐❡s ♦❢ S1✿ S2 ≤ S1❀ [S : S1] = [S1 : S2] = p❀ ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ ❀ ❛♥❞ ❢♦r ❡✈❡r② ✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ✐s t❤❡ ❣r♦✉♣ ✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ ✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn S = Zn−1(S) Sn−1 = Z(S) Sn−2 = Z2(S) S3 = Zn−3(S) S2 = Zn−2(S) S1 S4 = Zn−4(S)
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t S2 = [S, S]✱ Si = [Si−1, S] ❢♦r ❡✈❡r② i ≥ 3 ❛♥❞ S1 = CS(S2/S4). Pr♦♣❡rt✐❡s ♦❢ S1✿ S2 ≤ S1❀ [S : S1] = [S1 : S2] = p❀ S1 ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ S❀ ❛♥❞ ❢♦r ❡✈❡r② ✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ✐s t❤❡ ❣r♦✉♣ ✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ ✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn S = Zn−1(S) Sn−1 = Z(S) Sn−2 = Z2(S) S3 = Zn−3(S) S2 = Zn−2(S) S1 S4 = Zn−4(S)
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t S2 = [S, S]✱ Si = [Si−1, S] ❢♦r ❡✈❡r② i ≥ 3 ❛♥❞ S1 = CS(S2/S4). Pr♦♣❡rt✐❡s ♦❢ S1✿ S2 ≤ S1❀ [S : S1] = [S1 : S2] = p❀ S1 ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ S❀ ❛♥❞ S1 = CS(Si/Si+2) ❢♦r ❡✈❡r② 2 ≤ i ≤ n − 3✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ✐s t❤❡ ❣r♦✉♣ ✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ ✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn S = Zn−1(S) Sn−1 = Z(S) Sn−2 = Z2(S) S3 = Zn−3(S) S2 = Zn−2(S) S1 S4 = Zn−4(S) CS(Z2(S))
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t S2 = [S, S]✱ Si = [Si−1, S] ❢♦r ❡✈❡r② i ≥ 3 ❛♥❞ S1 = CS(S2/S4). Pr♦♣❡rt✐❡s ♦❢ S1✿ S2 ≤ S1❀ [S : S1] = [S1 : S2] = p❀ S1 ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ S❀ ❛♥❞ S1 = CS(Si/Si+2) ❢♦r ❡✈❡r② 2 ≤ i ≤ n − 3✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ✐s t❤❡ ❣r♦✉♣ ✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ ✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn S = Zn−1(S) Sn−1 = Z(S) Sn−2 = Z2(S) S3 = Zn−3(S) S2 = Zn−2(S) S1 S4 = Zn−4(S) CS(Z2(S))
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t S2 = [S, S]✱ Si = [Si−1, S] ❢♦r ❡✈❡r② i ≥ 3 ❛♥❞ S1 = CS(S2/S4). Pr♦♣❡rt✐❡s ♦❢ S1✿ S2 ≤ S1❀ [S : S1] = [S1 : S2] = p❀ S1 ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ S❀ ❛♥❞ S1 = CS(Si/Si+2) ❢♦r ❡✈❡r② 2 ≤ i ≤ n − 3✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ S ✐s t❤❡ ❣r♦✉♣ CS(Sn−2) = CS(Z2(S))✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ S1✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn S = Zn−1(S) Sn−1 = Z(S) Sn−2 = Z2(S) S3 = Zn−3(S) S2 = Zn−2(S) S1 S4 = Zn−4(S) M CS(Z2(S))
▲❡t S ❜❡ ❛ p✲❣r♦✉♣ ❤❛✈✐♥❣ ♦r❞❡r pn ❛♥❞ ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✭✐✳❡✳ ❝❧❛ss n−1✮✳ ❙❡t S2 = [S, S]✱ Si = [Si−1, S] ❢♦r ❡✈❡r② i ≥ 3 ❛♥❞ S1 = CS(S2/S4). Pr♦♣❡rt✐❡s ♦❢ S1✿ S2 ≤ S1❀ [S : S1] = [S1 : S2] = p❀ S1 ✐s ❝❤❛r❛❝t❡r✐st✐❝ ✐♥ S❀ ❛♥❞ S1 = CS(Si/Si+2) ❢♦r ❡✈❡r② 2 ≤ i ≤ n − 3✳ ❆♥♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ S ✐s t❤❡ ❣r♦✉♣ CS(Sn−2) = CS(Z2(S))✱ t❤❛t ♠✐❣❤t ❝♦✐♥❝✐❞❡ ✇✐t❤ S1✳ ❊✈❡r② ♦t❤❡r ♠❛①✐♠❛❧ s✉❜❣r♦✉♣ ♦❢ S ❤❛s ♠❛①✐♠❛❧ ♥✐❧♣♦t❡♥❝② ❝❧❛ss✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✽ ✴ ✶✺
1 = Sn Z(S) Z2(S) S3 S2 S S1 Z3(S) M CS(E) = E NS(E) NS(NS(E))
❙✉♣♣♦s❡ p ✐s ♦❞❞ ❛♥❞ E ∼ = Cp × Cp ✐s ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ t❤❡ p✲❣r♦✉♣ S✳ ❚❤❡♥✿ Pr♦♣❡rt② ✶✿ CS(E) = E❀ Pr♦♣❡rt② ✷✿ t❤❡r❡ ❡①✐sts ❛ ♥♦♥✲tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✭ ✮ ♥♦r♠❛❧✐③✐♥❣ s✉❝❤ t❤❛t ❢♦r s♦♠❡ ❤❛✈✐♥❣ ♦r❞❡r ✳ ❙♦ ✐❢ ✱ ✇✐t❤ ✱ t❤❡♥ ❛♥❞
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✾ ✴ ✶✺
1 = Sn Z(S) Z2(S) S3 S2 S S1 Z3(S) M CS(E) = E NS(E) NS(NS(E)) λ−1 λ
❙✉♣♣♦s❡ p ✐s ♦❞❞ ❛♥❞ E ∼ = Cp × Cp ✐s ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ t❤❡ p✲❣r♦✉♣ S✳ ❚❤❡♥✿ Pr♦♣❡rt② ✶✿ CS(E) = E❀ Pr♦♣❡rt② ✷✿ t❤❡r❡ ❡①✐sts ❛ ♥♦♥✲tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ϕ ♦❢ S ✭ϕ ∈ AutF(S))✮ ♥♦r♠❛❧✐③✐♥❣ E s✉❝❤ t❤❛t ϕ|E = λ−1 λ
❢♦r s♦♠❡ λ ∈ GF(p) ❤❛✈✐♥❣ ♦r❞❡r p − 1✳ ❙♦ ✐❢ ✱ ✇✐t❤ ✱ t❤❡♥ ❛♥❞
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✾ ✴ ✶✺
1 = Sn Z(S) Z2(S) S3 S2 S S1 Z3(S) M CS(E) = E NS(E) NS(NS(E)) λ−1 λ
❙✉♣♣♦s❡ p ✐s ♦❞❞ ❛♥❞ E ∼ = Cp × Cp ✐s ❛♥ ❡ss❡♥t✐❛❧ s✉❜❣r♦✉♣ ♦❢ t❤❡ p✲❣r♦✉♣ S✳ ❚❤❡♥✿ Pr♦♣❡rt② ✶✿ CS(E) = E❀ Pr♦♣❡rt② ✷✿ t❤❡r❡ ❡①✐sts ❛ ♥♦♥✲tr✐✈✐❛❧ ❛✉t♦♠♦r♣❤✐s♠ ϕ ♦❢ S ✭ϕ ∈ AutF(S))✮ ♥♦r♠❛❧✐③✐♥❣ E s✉❝❤ t❤❛t ϕ|E = λ−1 λ
❢♦r s♦♠❡ λ ∈ GF(p) ❤❛✈✐♥❣ ♦r❞❡r p − 1✳ ❙♦ ✐❢ E = e × z✱ ✇✐t❤ z ∈ Z(S)✱ t❤❡♥ eϕ = eλ−1 ❛♥❞ zϕ = zλ.
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✾ ✴ ✶✺
1 = Sn Z(S) Z2(S) S3 S2 S S1 Z3(S) M CS(E) = E NS(E) NS(NS(E)) λ−1 λ
❲❡ ❝❛♥ ♣r♦✈❡ t❤❛t E S1 ❛♥❞ E CS(Z2(S)). ■♥ ♣❛rt✐❝✉❧❛r✱ r❡❝❛❧❧✐♥❣ t❤❛t ❢♦r ❡✈❡r② ✱ ✇❡ ❣❡t t❤❛t ❢♦r ❡✈❡r② ✳ ❚❤✐s ❢❛❝t ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♠♠✉t❛✲ t♦rs ❡♥❛❜❧❡ ✉s t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥ ❡✈❡r② q✉♦t✐❡♥t ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✵ ✴ ✶✺
1 = Sn Z(S) Z2(S) S3 S2 S S1 Z3(S) M CS(E) = E NS(E) NS(NS(E)) λ−1 λ
❲❡ ❝❛♥ ♣r♦✈❡ t❤❛t E S1 ❛♥❞ E CS(Z2(S)). ■♥ ♣❛rt✐❝✉❧❛r✱ r❡❝❛❧❧✐♥❣ t❤❛t S1 = CS(Si/Si+2) ❢♦r ❡✈❡r② 2 ≤ i ≤ n − 3✱ ✇❡ ❣❡t t❤❛t [E, Si] Si+2 ❢♦r ❡✈❡r② i ≥ 1✳ ❚❤✐s ❢❛❝t ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♠♠✉t❛✲ t♦rs ❡♥❛❜❧❡ ✉s t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❛❝t✐♦♥ ♦❢ ♦♥ ❡✈❡r② q✉♦t✐❡♥t ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✵ ✴ ✶✺
1 = Sn Z(S) Z2(S) S3 S2 S S1 Z3(S) M CS(E) = E NS(E) NS(NS(E)) λ−1 λ λ2 λ3 λn−2 λn−1 λ−1
❲❡ ❝❛♥ ♣r♦✈❡ t❤❛t E S1 ❛♥❞ E CS(Z2(S)). ■♥ ♣❛rt✐❝✉❧❛r✱ r❡❝❛❧❧✐♥❣ t❤❛t S1 = CS(Si/Si+2) ❢♦r ❡✈❡r② 2 ≤ i ≤ n − 3✱ ✇❡ ❣❡t t❤❛t [E, Si] Si+2 ❢♦r ❡✈❡r② i ≥ 1✳ ❚❤✐s ❢❛❝t ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♠♠✉t❛✲ t♦rs ❡♥❛❜❧❡ ✉s t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❛❝t✐♦♥ ♦❢ ϕ ♦♥ ❡✈❡r② q✉♦t✐❡♥t Si/Si+1✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✵ ✴ ✶✺
▲❡t p ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S ❝♦♥t❛✐♥✐♥❣ ❛ ♣❡❛r❧✳
❚❤❡♦r❡♠ ✷ ✭●✳✱ ✷✵✶✼✮
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ❛♥❞ ♦r❞❡r |S| ≥ p4✳ ❚❤❡♥ ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ✱ ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❛♥❞ ✱ ✐❢ r❡❞✉❝❡❞✱ ✐s ❦♥♦✇♥ ✭❈r❛✈❡♥✱ ❖❧✐✈❡r✱ ❙❡♠❡r❛r♦✮❀ ❛♥❞ ❀ ✱ ✱ ❤❛s ❡①♣♦♥❡♥t ❛♥❞ ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✶ ✴ ✶✺
▲❡t p ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S ❝♦♥t❛✐♥✐♥❣ ❛ ♣❡❛r❧✳
❚❤❡♦r❡♠ ✷ ✭●✳✱ ✷✵✶✼✮
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ❛♥❞ ♦r❞❡r |S| ≥ p4✳ ❚❤❡♥ p ≥ k ≥ 2 ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = pk+1✱ S1 ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❛♥❞ F✱ ✐❢ r❡❞✉❝❡❞✱ ✐s ❦♥♦✇♥ ✭❈r❛✈❡♥✱ ❖❧✐✈❡r✱ ❙❡♠❡r❛r♦✮❀ ❛♥❞ ❀ ✱ ✱ ❤❛s ❡①♣♦♥❡♥t ❛♥❞ ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✶ ✴ ✶✺
▲❡t p ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S ❝♦♥t❛✐♥✐♥❣ ❛ ♣❡❛r❧✳
❚❤❡♦r❡♠ ✷ ✭●✳✱ ✷✵✶✼✮
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ❛♥❞ ♦r❞❡r |S| ≥ p4✳ ❚❤❡♥ p ≥ k ≥ 2 ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = pk+1✱ S1 ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❛♥❞ F✱ ✐❢ r❡❞✉❝❡❞✱ ✐s ❦♥♦✇♥ ✭❈r❛✈❡♥✱ ❖❧✐✈❡r✱ ❙❡♠❡r❛r♦✮❀ k = p − 1 ❛♥❞ |S| ≥ pp+1❀ ✱ ✱ ❤❛s ❡①♣♦♥❡♥t ❛♥❞ ✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✶ ✴ ✶✺
▲❡t p ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S ❝♦♥t❛✐♥✐♥❣ ❛ ♣❡❛r❧✳
❚❤❡♦r❡♠ ✷ ✭●✳✱ ✷✵✶✼✮
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ❛♥❞ ♦r❞❡r |S| ≥ p4✳ ❚❤❡♥ p ≥ k ≥ 2 ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = pk+1✱ S1 ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❛♥❞ F✱ ✐❢ r❡❞✉❝❡❞✱ ✐s ❦♥♦✇♥ ✭❈r❛✈❡♥✱ ❖❧✐✈❡r✱ ❙❡♠❡r❛r♦✮❀ k = p − 1 ❛♥❞ |S| ≥ pp+1❀ k ≥ 3✱ k + 3 ≤ p ≤ 2k + 1✱ S ❤❛s ❡①♣♦♥❡♥t p ❛♥❞ |S| ≤ pp−1✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✶ ✴ ✶✺
1 = Sn Z(S) Z2(S) S2 S = Zn−1(S) S1 Z3(S) M E λ−1 λ [x, y] λ2 λ3 y Z4(S) λ4 x C
◆♦t❡ t❤❛t t❤❡ ❣r♦✉♣ Z3(S) ✐s ❛❧✇❛②s ❛❜❡❧✐❛♥ ✭✐t ❝❡♥tr❛❧✐③❡s Z2(S)✮✳ ❙✉♣♣♦s❡ ✐s ◆❖❚ ❛❜❡❧✐❛♥✳ ❚❤❡♥ ❛♥❞ ✳ ❙♦ t❤❡r❡ ❡①✐sts ❛♥❞ s✉❝❤ t❤❛t ✳ ❚❤✉s ❙✐♥❝❡ ❤❛s ♦r❞❡r ✱ t❤✐s ✐♠♣❧✐❡s ♦r
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✷ ✴ ✶✺
1 = Sn Z(S) Z2(S) S2 S = Zn−1(S) S1 Z3(S) M E λ−1 λ [x, y] λ2 λ3 y Z4(S) λ4 x C
◆♦t❡ t❤❛t t❤❡ ❣r♦✉♣ Z3(S) ✐s ❛❧✇❛②s ❛❜❡❧✐❛♥ ✭✐t ❝❡♥tr❛❧✐③❡s Z2(S)✮✳ ❙✉♣♣♦s❡ Z4(S) ✐s ◆❖❚ ❛❜❡❧✐❛♥✳ ❚❤❡♥ ❛♥❞ ✳ ❙♦ t❤❡r❡ ❡①✐sts ❛♥❞ s✉❝❤ t❤❛t ✳ ❚❤✉s ❙✐♥❝❡ ❤❛s ♦r❞❡r ✱ t❤✐s ✐♠♣❧✐❡s ♦r
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✷ ✴ ✶✺
1 = Sn Z(S) Z2(S) S2 S = Zn−1(S) S1 Z3(S) M E λ−1 λ [x, y] λ2 λ3 y Z4(S) λ4 x C
◆♦t❡ t❤❛t t❤❡ ❣r♦✉♣ Z3(S) ✐s ❛❧✇❛②s ❛❜❡❧✐❛♥ ✭✐t ❝❡♥tr❛❧✐③❡s Z2(S)✮✳ ❙✉♣♣♦s❡ Z4(S) ✐s ◆❖❚ ❛❜❡❧✐❛♥✳ ❚❤❡♥ |S| ≤ p7 ❛♥❞ [Z4(S), Z3(S)] = Z(S)✳ ❙♦ t❤❡r❡ ❡①✐sts ❛♥❞ s✉❝❤ t❤❛t ✳ ❚❤✉s ❙✐♥❝❡ ❤❛s ♦r❞❡r ✱ t❤✐s ✐♠♣❧✐❡s ♦r
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✷ ✴ ✶✺
1 = Sn Z(S) Z2(S) S2 S = Zn−1(S) S1 Z3(S) M E λ−1 λ [x, y] λ2 λ3 y Z4(S) λ4 x C
◆♦t❡ t❤❛t t❤❡ ❣r♦✉♣ Z3(S) ✐s ❛❧✇❛②s ❛❜❡❧✐❛♥ ✭✐t ❝❡♥tr❛❧✐③❡s Z2(S)✮✳ ❙✉♣♣♦s❡ Z4(S) ✐s ◆❖❚ ❛❜❡❧✐❛♥✳ ❚❤❡♥ |S| ≤ p7 ❛♥❞ [Z4(S), Z3(S)] = Z(S)✳ ❙♦ t❤❡r❡ ❡①✐sts x ∈ Z4(S) ❛♥❞ y ∈ Z3(S) s✉❝❤ t❤❛t 1 = [x, y] ∈ Z(S)✳ ❚❤✉s ❙✐♥❝❡ ❤❛s ♦r❞❡r ✱ t❤✐s ✐♠♣❧✐❡s ♦r
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✷ ✴ ✶✺
1 = Sn Z(S) Z2(S) S2 S = Zn−1(S) S1 Z3(S) M E λ−1 λ [x, y] λ2 λ3 y Z4(S) λ4 x C
◆♦t❡ t❤❛t t❤❡ ❣r♦✉♣ Z3(S) ✐s ❛❧✇❛②s ❛❜❡❧✐❛♥ ✭✐t ❝❡♥tr❛❧✐③❡s Z2(S)✮✳ ❙✉♣♣♦s❡ Z4(S) ✐s ◆❖❚ ❛❜❡❧✐❛♥✳ ❚❤❡♥ |S| ≤ p7 ❛♥❞ [Z4(S), Z3(S)] = Z(S)✳ ❙♦ t❤❡r❡ ❡①✐sts x ∈ Z4(S) ❛♥❞ y ∈ Z3(S) s✉❝❤ t❤❛t 1 = [x, y] ∈ Z(S)✳ ❚❤✉s [x, y]λ = [x, y]ϕ = [xλ4, yλ3] = [x, y]λ7. ❙✐♥❝❡ λ ❤❛s ♦r❞❡r p − 1✱ t❤✐s ✐♠♣❧✐❡s p = 3 ♦r p = 7.
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✷ ✴ ✶✺
▲❡t p ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S ❝♦♥t❛✐♥✐♥❣ ❛ ♣❡❛r❧✳
❚❤❡♦r❡♠ ✷ ✭●✳✱ ✷✵✶✼✮
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ❛♥❞ ♦r❞❡r |S| ≥ p4✳ ❚❤❡♥ p ≥ k ≥ 2 ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = pk+1 ❛♥❞ S1 ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥❀ k = p − 1 ❛♥❞ |S| ≥ pp+1❀ k ≥ 3✱ k + 3 ≤ p ≤ 2k + 1✱ S ❤❛s ❡①♣♦♥❡♥t p ❛♥❞ |S| ≤ pp−1✳
❈♦r♦❧❧❛r②
❙✉♣♣♦s❡ t❤❛t ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ✳ ❚❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ❛♥❞ ❙②❧ ❀ ✭✐♠♣♦ss✐❜❧❡✮❀ ❛♥❞ ❙♠❛❧❧●r♦✉♣✭✼❫✺✱✸✼✮ ✭❤❛s ♦r❞❡r ❛♥❞ ❡①♣♦♥❡♥t ✮✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✸ ✴ ✶✺
▲❡t p ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S ❝♦♥t❛✐♥✐♥❣ ❛ ♣❡❛r❧✳
❚❤❡♦r❡♠ ✷ ✭●✳✱ ✷✵✶✼✮
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ❛♥❞ ♦r❞❡r |S| ≥ p4✳ ❚❤❡♥ p ≥ k ≥ 2 ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = pk+1 ❛♥❞ S1 ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥❀ k = p − 1 ❛♥❞ |S| ≥ pp+1❀ k ≥ 3✱ k + 3 ≤ p ≤ 2k + 1✱ S ❤❛s ❡①♣♦♥❡♥t p ❛♥❞ |S| ≤ pp−1✳
❈♦r♦❧❧❛r②
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k = 3✳ ❚❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = p4 ❛♥❞ S ∈ ❙②❧p(Sp4(p))❀ ✭✐♠♣♦ss✐❜❧❡✮❀ ❛♥❞ ❙♠❛❧❧●r♦✉♣✭✼❫✺✱✸✼✮ ✭❤❛s ♦r❞❡r ❛♥❞ ❡①♣♦♥❡♥t ✮✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✸ ✴ ✶✺
▲❡t p ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S ❝♦♥t❛✐♥✐♥❣ ❛ ♣❡❛r❧✳
❚❤❡♦r❡♠ ✷ ✭●✳✱ ✷✵✶✼✮
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ❛♥❞ ♦r❞❡r |S| ≥ p4✳ ❚❤❡♥ p ≥ k ≥ 2 ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = pk+1 ❛♥❞ S1 ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥❀ k = p − 1 ❛♥❞ |S| ≥ pp+1❀ k ≥ 3✱ k + 3 ≤ p ≤ 2k + 1✱ S ❤❛s ❡①♣♦♥❡♥t p ❛♥❞ |S| ≤ pp−1✳
❈♦r♦❧❧❛r②
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k = 3✳ ❚❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = p4 ❛♥❞ S ∈ ❙②❧p(Sp4(p))❀ 3 = p − 1 ✭✐♠♣♦ss✐❜❧❡✮❀ ❛♥❞ ❙♠❛❧❧●r♦✉♣✭✼❫✺✱✸✼✮ ✭❤❛s ♦r❞❡r ❛♥❞ ❡①♣♦♥❡♥t ✮✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✸ ✴ ✶✺
▲❡t p ❜❡ ❛♥ ♦❞❞ ♣r✐♠❡ ❛♥❞ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S ❝♦♥t❛✐♥✐♥❣ ❛ ♣❡❛r❧✳
❚❤❡♦r❡♠ ✷ ✭●✳✱ ✷✵✶✼✮
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k ❛♥❞ ♦r❞❡r |S| ≥ p4✳ ❚❤❡♥ p ≥ k ≥ 2 ❛♥❞ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = pk+1 ❛♥❞ S1 ✐s ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥❀ k = p − 1 ❛♥❞ |S| ≥ pp+1❀ k ≥ 3✱ k + 3 ≤ p ≤ 2k + 1✱ S ❤❛s ❡①♣♦♥❡♥t p ❛♥❞ |S| ≤ pp−1✳
❈♦r♦❧❧❛r②
❙✉♣♣♦s❡ t❤❛t S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ k = 3✳ ❚❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = p4 ❛♥❞ S ∈ ❙②❧p(Sp4(p))❀ 3 = p − 1 ✭✐♠♣♦ss✐❜❧❡✮❀ p = 7 ❛♥❞ S ∼ =❙♠❛❧❧●r♦✉♣✭✼❫✺✱✸✼✮ ✭❤❛s ♦r❞❡r 75 ❛♥❞ ❡①♣♦♥❡♥t 7✮✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✸ ✴ ✶✺
❚❤❡♦r❡♠ ✸ ✭●✳✱ ✷✵✶✼✮
▲❡t p ≥ 5 ❜❡ ❛ ♣r✐♠❡✱ ❧❡t F ❜❡ ❛ s❛t✉r❛t❡❞ ❢✉s✐♦♥ s②st❡♠ ♦♥ t❤❡ p✲❣r♦✉♣ S✳ ❙✉♣♣♦s❡ t❤❛t Op(F) = 1 ❛♥❞ S ❤❛s s❡❝t✐♦♥❛❧ r❛♥❦ ✸✳ ❚❤❡♥ F ❝♦♥t❛✐♥s ❛ ♣❡❛r❧ ❛♥❞ s♦ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ |S| = p4 ❛♥❞ S ∈ ❙②❧p(Sp4(p))❀ p = 7✱ S ∼ =❙♠❛❧❧●r♦✉♣✭✼❫✺✱✸✼✮ ✭❤❛s ♦r❞❡r 75 ❛♥❞ ❡①♣♦♥❡♥t 7✮✱ F = AutF(S), AutF(E)✱ ✇❤❡r❡ E ∼ = C7 × C7 ✐s ❛♥ ❛❜❡❧✐❛♥ ♣❡❛r❧✱ ❛♥❞ F ✐s s✐♠♣❧❡ ❛♥❞ ❡①♦t✐❝✳
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✹ ✴ ✶✺
❱❛❧❡♥t✐♥❛ ●r❛③✐❛♥ ❋✉s✐♦♥ s②st❡♠s ❝♦♥t❛✐♥✐♥❣ ♣❡❛r❧s
✶✺ ✴ ✶✺