r trss t t - - PowerPoint PPT Presentation

❈♦♣r✐♠❡ ❛✉t♦♠♦r♣❤✐s♠s ❛❝t✐♥❣ ✇✐t❤ ♥✐❧♣♦t❡♥t ❝❡♥tr❛❧✐③❡rs

❊♠❡rs♦♥ ❋✳ ❞❡ ▼❡❧♦

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ✲ ❙✉♣♣♦rt❡❞ ❜② ❋❆P✲❉❋

• r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼

❋✐①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣

❉❡✜♥✐t✐♦♥

◮ ▲❡t ϕ ❜❡ ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❣r♦✉♣ G✳ ❲❡ ❞❡♥♦t❡ ❜② CG(ϕ) t❤❡

✜①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣ ♦❢ ϕ ✐♥ G✱ CG(ϕ) = {x ∈ G; xϕ = x}. ❚❤✐s s✉❜❣r♦✉♣ ✐s ❛❧s♦ ❝❛❧❧❡❞ ✏t❤❡ ❝❡♥tr❛❧✐③❡r ♦❢ ✐♥ ✑✳ ■❢ ✶✱ ✇❡ s❛② t❤❛t ✐s ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✳ ❙✐♠✐❧❛r❧②✱ ✐❢ ✇❡ ❞❡♥♦t❡ ❜② t❤❡ s✉❜❣r♦✉♣ ❢♦r ❛❧❧ ■❢ ✶✱ ✇❡ s❛② t❤❛t ✐s ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ✳

❋✐①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣

❉❡✜♥✐t✐♦♥

◮ ▲❡t ϕ ❜❡ ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❣r♦✉♣ G✳ ❲❡ ❞❡♥♦t❡ ❜② CG(ϕ) t❤❡

✜①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣ ♦❢ ϕ ✐♥ G✱ CG(ϕ) = {x ∈ G; xϕ = x}. ❚❤✐s s✉❜❣r♦✉♣ ✐s ❛❧s♦ ❝❛❧❧❡❞ ✏t❤❡ ❝❡♥tr❛❧✐③❡r ♦❢ ϕ ✐♥ G✑✳ ■❢ CG(ϕ) = ✶✱ ✇❡ s❛② t❤❛t ϕ ✐s ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ G✳ ❙✐♠✐❧❛r❧②✱ ✐❢ ✇❡ ❞❡♥♦t❡ ❜② t❤❡ s✉❜❣r♦✉♣ ❢♦r ❛❧❧ ■❢ ✶✱ ✇❡ s❛② t❤❛t ✐s ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ✳

❋✐①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣

❉❡✜♥✐t✐♦♥

◮ ▲❡t ϕ ❜❡ ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❣r♦✉♣ G✳ ❲❡ ❞❡♥♦t❡ ❜② CG(ϕ) t❤❡

✜①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣ ♦❢ ϕ ✐♥ G✱ CG(ϕ) = {x ∈ G; xϕ = x}. ❚❤✐s s✉❜❣r♦✉♣ ✐s ❛❧s♦ ❝❛❧❧❡❞ ✏t❤❡ ❝❡♥tr❛❧✐③❡r ♦❢ ϕ ✐♥ G✑✳ ■❢ CG(ϕ) = ✶✱ ✇❡ s❛② t❤❛t ϕ ✐s ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ G✳

◮ ❙✐♠✐❧❛r❧②✱ ✐❢ A ≤ Aut(G) ✇❡ ❞❡♥♦t❡ ❜② CG(A) t❤❡ s✉❜❣r♦✉♣

CG(A) = {x ∈ G; xa = x ❢♦r ❛❧❧ a ∈ A}. ■❢ CG(A) = ✶✱ ✇❡ s❛② t❤❛t A ✐s ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ G✳

❙♦♠❡ ✉s❡❢✉❧ r❡s✉❧ts ✐♥ ❝♦♣r✐♠❡ ❛❝t✐♦♥s✳ ■❢ ✐s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ ❛♥❞ ✶✱ t❤❡♥ ❢♦r ❛♥② ✲✐♥✈❛r✐❛♥t ♥♦r♠❛❧ s✉❜❣r♦✉♣ ♦❢ ✳ ■❢ ✐s ❛ ♥♦♥✲❝②❝❧✐❝ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ ❛♥❞ ✶✱ t❤❡♥ ✳

❙♦♠❡ ✉s❡❢✉❧ r❡s✉❧ts ✐♥ ❝♦♣r✐♠❡ ❛❝t✐♦♥s✳

◮ ■❢ A ✐s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛♥❞

(|A|, |G|) = ✶✱ t❤❡♥ CG/N(A) = CG(A)N/N ❢♦r ❛♥② A✲✐♥✈❛r✐❛♥t ♥♦r♠❛❧ s✉❜❣r♦✉♣ N ♦❢ G✳ ■❢ ✐s ❛ ♥♦♥✲❝②❝❧✐❝ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ ❛♥❞ ✶✱ t❤❡♥ ✳

❙♦♠❡ ✉s❡❢✉❧ r❡s✉❧ts ✐♥ ❝♦♣r✐♠❡ ❛❝t✐♦♥s✳

◮ ■❢ A ✐s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛♥❞

(|A|, |G|) = ✶✱ t❤❡♥ CG/N(A) = CG(A)N/N ❢♦r ❛♥② A✲✐♥✈❛r✐❛♥t ♥♦r♠❛❧ s✉❜❣r♦✉♣ N ♦❢ G✳

◮ ■❢ A ✐s ❛ ♥♦♥✲❝②❝❧✐❝ ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣

G ❛♥❞ (|A|, |G|) = ✶✱ t❤❡♥ G = CG(a) ; a ∈ A#✳

❲❡❧❧✲❦♥♦✇♥ r❡s✉❧ts✳✳✳

◮ ✭●✳ ❍✐❣♠❛♥✲✶✾✺✼✮ ■❢ ❛ ✜♥✐t❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣ G ❛❞♠✐ts ❛

✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r p✱ t❤❡♥ ✐ts ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s p✲❜♦✉♥❞❡❞✳ ✭❏✳ ❚❤♦♠♣s♦♥✲✶✾✺✾✮ ■❢ ❛ ✜♥✐t❡ ❣r♦✉♣ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r✱ t❤❡♥ ✐s ♥✐❧♣♦t❡♥t✳ ✭❆✳ ❚✉r✉❧❧✲✶✾✽✹✮ ■❢ ❛ ✜♥✐t❡ ❣r♦✉♣ ❛❞♠✐ts ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r s✉❝❤ t❤❛t ✐s ♥✐❧♣♦t❡♥t✱ t❤❡♥ ✐ts ❋✐tt✐♥❣ ❤❡✐❣❤t ✐s ❛t ♠♦st ✸✳ ✭❆✳ ❚✉r✉❧❧✲✶✾✽✹✮ ▲❡t ❜❡ ❛ ✜♥✐t❡ s♦❧✉❜❧❡ ❣r♦✉♣ ❛❞♠✐tt✐♥❣ ❛ s♦❧✉❜❧❡ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s s✉❝❤ t❤❛t ✶✳ ❚❤❡♥ ✷ ✱ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✇❤♦s❡ ♣r♦❞✉❝ts ❣✐✈❡s ✳

❲❡❧❧✲❦♥♦✇♥ r❡s✉❧ts✳✳✳

◮ ✭●✳ ❍✐❣♠❛♥✲✶✾✺✼✮ ■❢ ❛ ✜♥✐t❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣ G ❛❞♠✐ts ❛

✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r p✱ t❤❡♥ ✐ts ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s p✲❜♦✉♥❞❡❞✳

◮ ✭❏✳ ❚❤♦♠♣s♦♥✲✶✾✺✾✮ ■❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡

❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r✱ t❤❡♥ G ✐s ♥✐❧♣♦t❡♥t✳ ✭❆✳ ❚✉r✉❧❧✲✶✾✽✹✮ ■❢ ❛ ✜♥✐t❡ ❣r♦✉♣ ❛❞♠✐ts ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r s✉❝❤ t❤❛t ✐s ♥✐❧♣♦t❡♥t✱ t❤❡♥ ✐ts ❋✐tt✐♥❣ ❤❡✐❣❤t ✐s ❛t ♠♦st ✸✳ ✭❆✳ ❚✉r✉❧❧✲✶✾✽✹✮ ▲❡t ❜❡ ❛ ✜♥✐t❡ s♦❧✉❜❧❡ ❣r♦✉♣ ❛❞♠✐tt✐♥❣ ❛ s♦❧✉❜❧❡ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s s✉❝❤ t❤❛t ✶✳ ❚❤❡♥ ✷ ✱ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✇❤♦s❡ ♣r♦❞✉❝ts ❣✐✈❡s ✳

❲❡❧❧✲❦♥♦✇♥ r❡s✉❧ts✳✳✳

◮ ✭●✳ ❍✐❣♠❛♥✲✶✾✺✼✮ ■❢ ❛ ✜♥✐t❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣ G ❛❞♠✐ts ❛

✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r p✱ t❤❡♥ ✐ts ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s p✲❜♦✉♥❞❡❞✳

◮ ✭❏✳ ❚❤♦♠♣s♦♥✲✶✾✺✾✮ ■❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡

❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r✱ t❤❡♥ G ✐s ♥✐❧♣♦t❡♥t✳

◮ ✭❆✳ ❚✉r✉❧❧✲✶✾✽✹✮ ■❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛❞♠✐ts ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢

♣r✐♠❡ ♦r❞❡r s✉❝❤ t❤❛t CG(ϕ) ✐s ♥✐❧♣♦t❡♥t✱ t❤❡♥ ✐ts ❋✐tt✐♥❣ ❤❡✐❣❤t ✐s ❛t ♠♦st ✸✳ ✭❆✳ ❚✉r✉❧❧✲✶✾✽✹✮ ▲❡t ❜❡ ❛ ✜♥✐t❡ s♦❧✉❜❧❡ ❣r♦✉♣ ❛❞♠✐tt✐♥❣ ❛ s♦❧✉❜❧❡ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s s✉❝❤ t❤❛t ✶✳ ❚❤❡♥ ✷ ✱ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✇❤♦s❡ ♣r♦❞✉❝ts ❣✐✈❡s ✳

❲❡❧❧✲❦♥♦✇♥ r❡s✉❧ts✳✳✳

◮ ✭●✳ ❍✐❣♠❛♥✲✶✾✺✼✮ ■❢ ❛ ✜♥✐t❡ ♥✐❧♣♦t❡♥t ❣r♦✉♣ G ❛❞♠✐ts ❛

✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r p✱ t❤❡♥ ✐ts ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s p✲❜♦✉♥❞❡❞✳

◮ ✭❏✳ ❚❤♦♠♣s♦♥✲✶✾✺✾✮ ■❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡

❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♣r✐♠❡ ♦r❞❡r✱ t❤❡♥ G ✐s ♥✐❧♣♦t❡♥t✳

◮ ✭❆✳ ❚✉r✉❧❧✲✶✾✽✹✮ ■❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛❞♠✐ts ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢

♣r✐♠❡ ♦r❞❡r s✉❝❤ t❤❛t CG(ϕ) ✐s ♥✐❧♣♦t❡♥t✱ t❤❡♥ ✐ts ❋✐tt✐♥❣ ❤❡✐❣❤t ✐s ❛t ♠♦st ✸✳

◮ ✭❆✳ ❚✉r✉❧❧✲✶✾✽✹✮ ▲❡t G ❜❡ ❛ ✜♥✐t❡ s♦❧✉❜❧❡ ❣r♦✉♣ ❛❞♠✐tt✐♥❣ ❛ s♦❧✉❜❧❡

❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s A s✉❝❤ t❤❛t (|G|, |A|) = ✶✳ ❚❤❡♥ h(G) ≤ h(CG(A)) + ✷k(A)✱ ✇❤❡r❡ k ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✇❤♦s❡ ♣r♦❞✉❝ts ❣✐✈❡s |A|✳

◆♦♥✲❝②❝❧✐❝ ❝❛s❡✳✳✳ ✭❏✳ ◆✳ ❲❛r❞✲✶✾✻✾✮ ▲❡t ❜❡ ❛ ♣r✐♠❡✱ ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ✲❣r♦✉♣ ♦❢ ♦r❞❡r

✷ ❛♥❞

❛ ✜♥✐t❡ ✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t ❛❝ts ♦♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐s ♥✐❧♣♦t❡♥t ❢♦r ❛♥② ✳ ❚❤❡♥ ❤❛s ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣ s✉❝❤ t❤❛t ❜♦t❤ ❛♥❞ ❛r❡ ♥✐❧♣♦t❡♥t✳ ✭❏✳ ◆✳ ❲❛r❞✲✶✾✼✶✮ ▲❡t ❜❡ ❛ ♣r✐♠❡✱ ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ✲❣r♦✉♣ ♦❢ ♦r❞❡r

✸ ❛♥❞

❛ ✜♥✐t❡ ✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t ❛❝ts ♦♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐s ♥✐❧♣♦t❡♥t ❢♦r ❛♥② ✳ ❚❤❡♥ ✐s ♥✐❧♣♦t❡♥t✳ ✭P✳ ❙❤✉♠②❛ts❦②✲✷✵✵✶✮ ▲❡t ❜❡ ❛ ♣r✐♠❡✱ ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ✲❣r♦✉♣ ♦❢ ♦r❞❡r

✸ ❛♥❞

❛ ✜♥✐t❡ ✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t ❛❝ts ♦♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st ❢♦r ❛♥② ✳ ❚❤❡♥ ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ ❛♥❞ ✳

◆♦♥✲❝②❝❧✐❝ ❝❛s❡✳✳✳

◮ ✭❏✳ ◆✳ ❲❛r❞✲✶✾✻✾✮ ▲❡t p ❜❡ ❛ ♣r✐♠❡✱ A ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥

p✲❣r♦✉♣ ♦❢ ♦r❞❡r p✷ ❛♥❞ G ❛ ✜♥✐t❡ p′✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t A ❛❝ts ♦♥ G ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t CG(a) ✐s ♥✐❧♣♦t❡♥t ❢♦r ❛♥② a ∈ A#✳ ❚❤❡♥ G ❤❛s ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣ F s✉❝❤ t❤❛t ❜♦t❤ F ❛♥❞ G/F ❛r❡ ♥✐❧♣♦t❡♥t✳ ✭❏✳ ◆✳ ❲❛r❞✲✶✾✼✶✮ ▲❡t ❜❡ ❛ ♣r✐♠❡✱ ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ✲❣r♦✉♣ ♦❢ ♦r❞❡r

✸ ❛♥❞

❛ ✜♥✐t❡ ✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t ❛❝ts ♦♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐s ♥✐❧♣♦t❡♥t ❢♦r ❛♥② ✳ ❚❤❡♥ ✐s ♥✐❧♣♦t❡♥t✳ ✭P✳ ❙❤✉♠②❛ts❦②✲✷✵✵✶✮ ▲❡t ❜❡ ❛ ♣r✐♠❡✱ ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ✲❣r♦✉♣ ♦❢ ♦r❞❡r

✸ ❛♥❞

❛ ✜♥✐t❡ ✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t ❛❝ts ♦♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st ❢♦r ❛♥② ✳ ❚❤❡♥ ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ ❛♥❞ ✳

◆♦♥✲❝②❝❧✐❝ ❝❛s❡✳✳✳

◮ ✭❏✳ ◆✳ ❲❛r❞✲✶✾✻✾✮ ▲❡t p ❜❡ ❛ ♣r✐♠❡✱ A ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥

p✲❣r♦✉♣ ♦❢ ♦r❞❡r p✷ ❛♥❞ G ❛ ✜♥✐t❡ p′✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t A ❛❝ts ♦♥ G ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t CG(a) ✐s ♥✐❧♣♦t❡♥t ❢♦r ❛♥② a ∈ A#✳ ❚❤❡♥ G ❤❛s ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣ F s✉❝❤ t❤❛t ❜♦t❤ F ❛♥❞ G/F ❛r❡ ♥✐❧♣♦t❡♥t✳

◮ ✭❏✳ ◆✳ ❲❛r❞✲✶✾✼✶✮ ▲❡t p ❜❡ ❛ ♣r✐♠❡✱ A ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥

p✲❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ G ❛ ✜♥✐t❡ p′✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t A ❛❝ts ♦♥ G ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t CG(a) ✐s ♥✐❧♣♦t❡♥t ❢♦r ❛♥② a ∈ A#✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t✳ ✭P✳ ❙❤✉♠②❛ts❦②✲✷✵✵✶✮ ▲❡t ❜❡ ❛ ♣r✐♠❡✱ ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ✲❣r♦✉♣ ♦❢ ♦r❞❡r

✸ ❛♥❞

❛ ✜♥✐t❡ ✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t ❛❝ts ♦♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st ❢♦r ❛♥② ✳ ❚❤❡♥ ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ ❛♥❞ ✳

◆♦♥✲❝②❝❧✐❝ ❝❛s❡✳✳✳

◮ ✭❏✳ ◆✳ ❲❛r❞✲✶✾✻✾✮ ▲❡t p ❜❡ ❛ ♣r✐♠❡✱ A ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥

p✲❣r♦✉♣ ♦❢ ♦r❞❡r p✷ ❛♥❞ G ❛ ✜♥✐t❡ p′✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t A ❛❝ts ♦♥ G ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t CG(a) ✐s ♥✐❧♣♦t❡♥t ❢♦r ❛♥② a ∈ A#✳ ❚❤❡♥ G ❤❛s ❛ ♥♦r♠❛❧ s✉❜❣r♦✉♣ F s✉❝❤ t❤❛t ❜♦t❤ F ❛♥❞ G/F ❛r❡ ♥✐❧♣♦t❡♥t✳

◮ ✭❏✳ ◆✳ ❲❛r❞✲✶✾✼✶✮ ▲❡t p ❜❡ ❛ ♣r✐♠❡✱ A ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥

p✲❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ G ❛ ✜♥✐t❡ p′✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t A ❛❝ts ♦♥ G ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t CG(a) ✐s ♥✐❧♣♦t❡♥t ❢♦r ❛♥② a ∈ A#✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t✳

◮ ✭P✳ ❙❤✉♠②❛ts❦②✲✷✵✵✶✮ ▲❡t p ❜❡ ❛ ♣r✐♠❡✱ A ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥

p✲❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ G ❛ ✜♥✐t❡ p′✲❣r♦✉♣✳ ❙✉♣♣♦s❡ t❤❛t A ❛❝ts ♦♥ G ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② a ∈ A#✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳

◆♦♥✲❛❜❡❧✐❛♥ ❝❛s❡

❚❤❡♦r❡♠ ✭❙❤✉♠②❛ts❦②✱ ❞❡ ▼❡❧♦ ✲ ✷✵✶✻✮

▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ A ❛ ✜♥✐t❡ ❣r♦✉♣ ♦❢ ❡①♣♦♥❡♥t p ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ p′✲❣r♦✉♣ G✳ ❆ss✉♠❡ t❤❛t A ❤❛s ♦r❞❡r ❛t ❧❡❛st p✸ ❛♥❞ CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ ❡❧❡♠❡♥t ♦❢ A✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t ✳ ❚❤❛t ✐s✱ ✐s ❛♥ ❡①tr❛✲s♣❡❝✐❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r

✸ ❛♥❞

✳ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ ✐s ❛❜❡❧✐❛♥ ❢♦r ❛♥② ✳ ❙❡t

✵

✳

✵ ✐s

✲✐♥✈❛r✐❛♥t ❛♥❞ ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦r❞❡r

✷✳ ❚❤✉s✱ ✵

✳ ■♥ ♦t❤❡r ✇♦r❞s✱

✵

❛♥❞

✷ ✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r

✷ ♦❢

✇❡ ❤❛✈❡ ✳ ❚❤❡♥

✵

✶ ❚❤❡r❡❢♦r❡✱ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r ✳

◆♦♥✲❛❜❡❧✐❛♥ ❝❛s❡

❚❤❡♦r❡♠ ✭❙❤✉♠②❛ts❦②✱ ❞❡ ▼❡❧♦ ✲ ✷✵✶✻✮

▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ A ❛ ✜♥✐t❡ ❣r♦✉♣ ♦❢ ❡①♣♦♥❡♥t p ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ p′✲❣r♦✉♣ G✳ ❆ss✉♠❡ t❤❛t A ❤❛s ♦r❞❡r ❛t ❧❡❛st p✸ ❛♥❞ CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ ❡❧❡♠❡♥t ♦❢ A✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳

◮ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t A = (Cp × Cp) ⋊ Cp✳ ❚❤❛t ✐s✱ A ✐s ❛♥

❡①tr❛✲s♣❡❝✐❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ |Z(A)| = p✳ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ ✐s ❛❜❡❧✐❛♥ ❢♦r ❛♥② ✳ ❙❡t

✵

✳

✵ ✐s

✲✐♥✈❛r✐❛♥t ❛♥❞ ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦r❞❡r

✷✳ ❚❤✉s✱ ✵

✳ ■♥ ♦t❤❡r ✇♦r❞s✱

✵

❛♥❞

✷ ✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r

✷ ♦❢

✇❡ ❤❛✈❡ ✳ ❚❤❡♥

✵

✶ ❚❤❡r❡❢♦r❡✱ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r ✳

◆♦♥✲❛❜❡❧✐❛♥ ❝❛s❡

❚❤❡♦r❡♠ ✭❙❤✉♠②❛ts❦②✱ ❞❡ ▼❡❧♦ ✲ ✷✵✶✻✮

▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ A ❛ ✜♥✐t❡ ❣r♦✉♣ ♦❢ ❡①♣♦♥❡♥t p ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ p′✲❣r♦✉♣ G✳ ❆ss✉♠❡ t❤❛t A ❤❛s ♦r❞❡r ❛t ❧❡❛st p✸ ❛♥❞ CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ ❡❧❡♠❡♥t ♦❢ A✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳

◮ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t A = (Cp × Cp) ⋊ Cp✳ ❚❤❛t ✐s✱ A ✐s ❛♥

❡①tr❛✲s♣❡❝✐❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ |Z(A)| = p✳

◮ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t G ✐s ♥✐❧♣♦t❡♥t✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t G ✐s

♥✐❧♣♦t❡♥t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ CG(a) ✐s ❛❜❡❧✐❛♥ ❢♦r ❛♥② a ∈ A#✳ ❙❡t

✵

✳

✵ ✐s

✲✐♥✈❛r✐❛♥t ❛♥❞ ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦r❞❡r

✷✳ ❚❤✉s✱ ✵

✳ ■♥ ♦t❤❡r ✇♦r❞s✱

✵

❛♥❞

✷ ✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r

✷ ♦❢

✇❡ ❤❛✈❡ ✳ ❚❤❡♥

✵

✶ ❚❤❡r❡❢♦r❡✱ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r ✳

◆♦♥✲❛❜❡❧✐❛♥ ❝❛s❡

❚❤❡♦r❡♠ ✭❙❤✉♠②❛ts❦②✱ ❞❡ ▼❡❧♦ ✲ ✷✵✶✻✮

▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ A ❛ ✜♥✐t❡ ❣r♦✉♣ ♦❢ ❡①♣♦♥❡♥t p ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ p′✲❣r♦✉♣ G✳ ❆ss✉♠❡ t❤❛t A ❤❛s ♦r❞❡r ❛t ❧❡❛st p✸ ❛♥❞ CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ ❡❧❡♠❡♥t ♦❢ A✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳

◮ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t A = (Cp × Cp) ⋊ Cp✳ ❚❤❛t ✐s✱ A ✐s ❛♥

❡①tr❛✲s♣❡❝✐❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ |Z(A)| = p✳

◮ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t G ✐s ♥✐❧♣♦t❡♥t✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t G ✐s

♥✐❧♣♦t❡♥t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ CG(a) ✐s ❛❜❡❧✐❛♥ ❢♦r ❛♥② a ∈ A#✳

◮ ❙❡t G✵ = CG(Z(A))✳ ✵ ✐s

✲✐♥✈❛r✐❛♥t ❛♥❞ ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢ ♦r❞❡r

✷✳ ❚❤✉s✱ ✵

✳ ■♥ ♦t❤❡r ✇♦r❞s✱

✵

❛♥❞

✷ ✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r

✷ ♦❢

✇❡ ❤❛✈❡ ✳ ❚❤❡♥

✵

✶ ❚❤❡r❡❢♦r❡✱ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r ✳

◆♦♥✲❛❜❡❧✐❛♥ ❝❛s❡

❚❤❡♦r❡♠ ✭❙❤✉♠②❛ts❦②✱ ❞❡ ▼❡❧♦ ✲ ✷✵✶✻✮

▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ A ❛ ✜♥✐t❡ ❣r♦✉♣ ♦❢ ❡①♣♦♥❡♥t p ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ p′✲❣r♦✉♣ G✳ ❆ss✉♠❡ t❤❛t A ❤❛s ♦r❞❡r ❛t ❧❡❛st p✸ ❛♥❞ CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ ❡❧❡♠❡♥t ♦❢ A✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳

◮ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t A = (Cp × Cp) ⋊ Cp✳ ❚❤❛t ✐s✱ A ✐s ❛♥

❡①tr❛✲s♣❡❝✐❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ |Z(A)| = p✳

◮ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t G ✐s ♥✐❧♣♦t❡♥t✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t G ✐s

♥✐❧♣♦t❡♥t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ CG(a) ✐s ❛❜❡❧✐❛♥ ❢♦r ❛♥② a ∈ A#✳

◮ ❙❡t G✵ = CG(Z(A))✳ ◮ G✵ ✐s A✲✐♥✈❛r✐❛♥t ❛♥❞ A = A/Z(A) ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢

♦r❞❡r p✷✳ ❚❤✉s✱ G✵ = CG(a) ; a ∈ A

#✳

■♥ ♦t❤❡r ✇♦r❞s✱

✵

❛♥❞

✷ ✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r

✷ ♦❢

✇❡ ❤❛✈❡ ✳ ❚❤❡♥

✵

✶ ❚❤❡r❡❢♦r❡✱ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r ✳

◆♦♥✲❛❜❡❧✐❛♥ ❝❛s❡

❚❤❡♦r❡♠ ✭❙❤✉♠②❛ts❦②✱ ❞❡ ▼❡❧♦ ✲ ✷✵✶✻✮

▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ A ❛ ✜♥✐t❡ ❣r♦✉♣ ♦❢ ❡①♣♦♥❡♥t p ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ p′✲❣r♦✉♣ G✳ ❆ss✉♠❡ t❤❛t A ❤❛s ♦r❞❡r ❛t ❧❡❛st p✸ ❛♥❞ CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ ❡❧❡♠❡♥t ♦❢ A✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳

◮ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t A = (Cp × Cp) ⋊ Cp✳ ❚❤❛t ✐s✱ A ✐s ❛♥

❡①tr❛✲s♣❡❝✐❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ |Z(A)| = p✳

◮ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t G ✐s ♥✐❧♣♦t❡♥t✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t G ✐s

♥✐❧♣♦t❡♥t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ CG(a) ✐s ❛❜❡❧✐❛♥ ❢♦r ❛♥② a ∈ A#✳

◮ ❙❡t G✵ = CG(Z(A))✳ ◮ G✵ ✐s A✲✐♥✈❛r✐❛♥t ❛♥❞ A = A/Z(A) ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢

♦r❞❡r p✷✳ ❚❤✉s✱ G✵ = CG(a) ; a ∈ A

#✳ ◮ ■♥ ♦t❤❡r ✇♦r❞s✱ G✵ = CG(B) ; B < A ❛♥❞ |B| = p✷✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② s✉❜❣r♦✉♣ ♦❢ ♦r❞❡r

✷ ♦❢

✇❡ ❤❛✈❡ ✳ ❚❤❡♥

✵

✶ ❚❤❡r❡❢♦r❡✱ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r ✳

◆♦♥✲❛❜❡❧✐❛♥ ❝❛s❡

❚❤❡♦r❡♠ ✭❙❤✉♠②❛ts❦②✱ ❞❡ ▼❡❧♦ ✲ ✷✵✶✻✮

▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ A ❛ ✜♥✐t❡ ❣r♦✉♣ ♦❢ ❡①♣♦♥❡♥t p ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ p′✲❣r♦✉♣ G✳ ❆ss✉♠❡ t❤❛t A ❤❛s ♦r❞❡r ❛t ❧❡❛st p✸ ❛♥❞ CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ ❡❧❡♠❡♥t ♦❢ A✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳

◮ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t A = (Cp × Cp) ⋊ Cp✳ ❚❤❛t ✐s✱ A ✐s ❛♥

❡①tr❛✲s♣❡❝✐❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ |Z(A)| = p✳

◮ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t G ✐s ♥✐❧♣♦t❡♥t✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t G ✐s

♥✐❧♣♦t❡♥t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ CG(a) ✐s ❛❜❡❧✐❛♥ ❢♦r ❛♥② a ∈ A#✳

◮ ❙❡t G✵ = CG(Z(A))✳ ◮ G✵ ✐s A✲✐♥✈❛r✐❛♥t ❛♥❞ A = A/Z(A) ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢

♦r❞❡r p✷✳ ❚❤✉s✱ G✵ = CG(a) ; a ∈ A

#✳ ◮ ■♥ ♦t❤❡r ✇♦r❞s✱ G✵ = CG(B) ; B < A ❛♥❞ |B| = p✷✳ ◮ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② s✉❜❣r♦✉♣B ♦❢ ♦r❞❡r p✷ ♦❢ A ✇❡ ❤❛✈❡

G = CG(b) ; b ∈ B✳ ❚❤❡♥ [G, G✵] = ✶. ❚❤❡r❡❢♦r❡✱ ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r ✳

◆♦♥✲❛❜❡❧✐❛♥ ❝❛s❡

❚❤❡♦r❡♠ ✭❙❤✉♠②❛ts❦②✱ ❞❡ ▼❡❧♦ ✲ ✷✵✶✻✮

▲❡t p ❜❡ ❛ ♣r✐♠❡ ❛♥❞ A ❛ ✜♥✐t❡ ❣r♦✉♣ ♦❢ ❡①♣♦♥❡♥t p ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ p′✲❣r♦✉♣ G✳ ❆ss✉♠❡ t❤❛t A ❤❛s ♦r❞❡r ❛t ❧❡❛st p✸ ❛♥❞ CG(a) ✐s ♥✐❧♣♦t❡♥t ♦❢ ❝❧❛ss ❛t ♠♦st c ❢♦r ❛♥② ♥♦♥tr✐✈✐❛❧ ❡❧❡♠❡♥t ♦❢ A✳ ❚❤❡♥ G ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ❝❧❛ss ❜♦✉♥❞❡❞ ✐♥ t❡r♠s ♦❢ c ❛♥❞ p✳

◮ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t A = (Cp × Cp) ⋊ Cp✳ ❚❤❛t ✐s✱ A ✐s ❛♥

❡①tr❛✲s♣❡❝✐❛❧ ❣r♦✉♣ ♦❢ ♦r❞❡r p✸ ❛♥❞ |Z(A)| = p✳

◮ ❋✐rst✱ ✇❡ ♣r♦✈❡ t❤❛t G ✐s ♥✐❧♣♦t❡♥t✳ ■t ✐s s✉✣❝✐❡♥t t♦ ♣r♦✈❡ t❤❛t G ✐s

♥✐❧♣♦t❡♥t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ CG(a) ✐s ❛❜❡❧✐❛♥ ❢♦r ❛♥② a ∈ A#✳

◮ ❙❡t G✵ = CG(Z(A))✳ ◮ G✵ ✐s A✲✐♥✈❛r✐❛♥t ❛♥❞ A = A/Z(A) ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❛❜❡❧✐❛♥ ❣r♦✉♣ ♦❢

♦r❞❡r p✷✳ ❚❤✉s✱ G✵ = CG(a) ; a ∈ A

#✳ ◮ ■♥ ♦t❤❡r ✇♦r❞s✱ G✵ = CG(B) ; B < A ❛♥❞ |B| = p✷✳ ◮ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r ❛♥② s✉❜❣r♦✉♣B ♦❢ ♦r❞❡r p✷ ♦❢ A ✇❡ ❤❛✈❡

G = CG(b) ; b ∈ B✳ ❚❤❡♥ [G, G✵] = ✶.

◮ ❚❤❡r❡❢♦r❡✱ G/Z(G) ❛❞♠✐ts ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r

p✳

◮ ◆♦✇✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s (c, p)✲❜♦✉♥❞❡❞✳

❈♦♥s✐❞❡r t❤❡ ❛ss♦❝✐❛t❡❞ ▲✐❡ r✐♥❣ ♦❢ t❤❡ ❣r♦✉♣

✶ ✶

✇❤❡r❡ ✐s t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ ❛♥❞ ❛r❡ t❤❡ t❡r♠s ♦❢ t❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ ✳ ❚❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ ✳ ❙❡t

✵

✳ ❚❤❡♥ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ ✲❜♦✉♥❞❡❞ ♥✉♠❜❡r s✉❝❤ t❤❛t

✵ ✵

✵✳ ❚❤✉s✱ t❤❡ ▲✐❡ r✐♥❣ ✐s s♦❧✉❜❧❡ ✇✐t❤ ✲❜♦✉♥❞❡❞ ❞❡r✐✈❡❞ ❧❡♥❣t❤✳ ◆♦✇✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ✐s ♠❡t❛❜❡❧✐❛♥ ❛♥❞ t❤❡♥ ✇❡ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ✲❜♦✉♥❞❡❞ ❝❧❛ss✳

◮ ◆♦✇✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s (c, p)✲❜♦✉♥❞❡❞✳ ◮ ❈♦♥s✐❞❡r t❤❡ ❛ss♦❝✐❛t❡❞ ▲✐❡ r✐♥❣ ♦❢ t❤❡ ❣r♦✉♣ G

L(G) =

n

• i=✶

γi/γi+✶, ✇❤❡r❡ n ✐s t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❛♥❞ γi ❛r❡ t❤❡ t❡r♠s ♦❢ t❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ G✳ ❚❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ L(G)✳ ❙❡t

✵

✳ ❚❤❡♥ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ ✲❜♦✉♥❞❡❞ ♥✉♠❜❡r s✉❝❤ t❤❛t

✵ ✵

✵✳ ❚❤✉s✱ t❤❡ ▲✐❡ r✐♥❣ ✐s s♦❧✉❜❧❡ ✇✐t❤ ✲❜♦✉♥❞❡❞ ❞❡r✐✈❡❞ ❧❡♥❣t❤✳ ◆♦✇✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ✐s ♠❡t❛❜❡❧✐❛♥ ❛♥❞ t❤❡♥ ✇❡ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ✲❜♦✉♥❞❡❞ ❝❧❛ss✳

◮ ◆♦✇✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s (c, p)✲❜♦✉♥❞❡❞✳ ◮ ❈♦♥s✐❞❡r t❤❡ ❛ss♦❝✐❛t❡❞ ▲✐❡ r✐♥❣ ♦❢ t❤❡ ❣r♦✉♣ G

L(G) =

n

• i=✶

γi/γi+✶, ✇❤❡r❡ n ✐s t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❛♥❞ γi ❛r❡ t❤❡ t❡r♠s ♦❢ t❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ G✳ ❚❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ L(G)✳

◮ ❙❡t L✵ = CL(Z(A))✳

❚❤❡♥ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ ✲❜♦✉♥❞❡❞ ♥✉♠❜❡r s✉❝❤ t❤❛t

✵ ✵

✵✳ ❚❤✉s✱ t❤❡ ▲✐❡ r✐♥❣ ✐s s♦❧✉❜❧❡ ✇✐t❤ ✲❜♦✉♥❞❡❞ ❞❡r✐✈❡❞ ❧❡♥❣t❤✳ ◆♦✇✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ✐s ♠❡t❛❜❡❧✐❛♥ ❛♥❞ t❤❡♥ ✇❡ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ✲❜♦✉♥❞❡❞ ❝❧❛ss✳

◮ ◆♦✇✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s (c, p)✲❜♦✉♥❞❡❞✳ ◮ ❈♦♥s✐❞❡r t❤❡ ❛ss♦❝✐❛t❡❞ ▲✐❡ r✐♥❣ ♦❢ t❤❡ ❣r♦✉♣ G

L(G) =

n

• i=✶

γi/γi+✶, ✇❤❡r❡ n ✐s t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❛♥❞ γi ❛r❡ t❤❡ t❡r♠s ♦❢ t❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ G✳ ❚❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ L(G)✳

◮ ❙❡t L✵ = CL(Z(A))✳ ◮ ❚❤❡♥ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ (c, p)✲❜♦✉♥❞❡❞ ♥✉♠❜❡r u

s✉❝❤ t❤❛t [L, L✵, . . . , L✵

• u

] = ✵✳ ❚❤✉s✱ t❤❡ ▲✐❡ r✐♥❣ ✐s s♦❧✉❜❧❡ ✇✐t❤ ✲❜♦✉♥❞❡❞ ❞❡r✐✈❡❞ ❧❡♥❣t❤✳ ◆♦✇✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ✐s ♠❡t❛❜❡❧✐❛♥ ❛♥❞ t❤❡♥ ✇❡ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ✲❜♦✉♥❞❡❞ ❝❧❛ss✳

◮ ◆♦✇✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s (c, p)✲❜♦✉♥❞❡❞✳ ◮ ❈♦♥s✐❞❡r t❤❡ ❛ss♦❝✐❛t❡❞ ▲✐❡ r✐♥❣ ♦❢ t❤❡ ❣r♦✉♣ G

L(G) =

n

• i=✶

γi/γi+✶, ✇❤❡r❡ n ✐s t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❛♥❞ γi ❛r❡ t❤❡ t❡r♠s ♦❢ t❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ G✳ ❚❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ L(G)✳

◮ ❙❡t L✵ = CL(Z(A))✳ ◮ ❚❤❡♥ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ (c, p)✲❜♦✉♥❞❡❞ ♥✉♠❜❡r u

s✉❝❤ t❤❛t [L, L✵, . . . , L✵

• u

] = ✵✳

◮ ❚❤✉s✱ t❤❡ ▲✐❡ r✐♥❣ L(G) ✐s s♦❧✉❜❧❡ ✇✐t❤ (c, p)✲❜♦✉♥❞❡❞ ❞❡r✐✈❡❞

❧❡♥❣t❤✳ ◆♦✇✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t ✐s ♠❡t❛❜❡❧✐❛♥ ❛♥❞ t❤❡♥ ✇❡ ♣r♦✈❡ t❤❛t ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ ✲❜♦✉♥❞❡❞ ❝❧❛ss✳

◮ ◆♦✇✱ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ✐s (c, p)✲❜♦✉♥❞❡❞✳ ◮ ❈♦♥s✐❞❡r t❤❡ ❛ss♦❝✐❛t❡❞ ▲✐❡ r✐♥❣ ♦❢ t❤❡ ❣r♦✉♣ G

L(G) =

n

• i=✶

γi/γi+✶, ✇❤❡r❡ n ✐s t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❛♥❞ γi ❛r❡ t❤❡ t❡r♠s ♦❢ t❤❡ ❧♦✇❡r ❝❡♥tr❛❧ s❡r✐❡s ♦❢ G✳ ❚❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ G ❝♦✐♥❝✐❞❡ ✇✐t❤ t❤❡ ♥✐❧♣♦t❡♥❝② ❝❧❛ss ♦❢ L(G)✳

◮ ❙❡t L✵ = CL(Z(A))✳ ◮ ❚❤❡♥ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ (c, p)✲❜♦✉♥❞❡❞ ♥✉♠❜❡r u

s✉❝❤ t❤❛t [L, L✵, . . . , L✵

• u

] = ✵✳

◮ ❚❤✉s✱ t❤❡ ▲✐❡ r✐♥❣ L(G) ✐s s♦❧✉❜❧❡ ✇✐t❤ (c, p)✲❜♦✉♥❞❡❞ ❞❡r✐✈❡❞

❧❡♥❣t❤✳

◮ ◆♦✇✱ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t L(G) ✐s ♠❡t❛❜❡❧✐❛♥ ❛♥❞ t❤❡♥ ✇❡ ♣r♦✈❡

t❤❛t L(G) ✐s ♥✐❧♣♦t❡♥t ✇✐t❤ (c, p)✲❜♦✉♥❞❡❞ ❝❧❛ss✳

❚❤❛♥❦ ②♦✉✦