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  1. ❈♦♣r✐♠❡ ❛✉t♦♠♦r♣❤✐s♠s ❛❝t✐♥❣ ✇✐t❤ ♥✐❧♣♦t❡♥t ❝❡♥tr❛❧✐③❡rs ❊♠❡rs♦♥ ❋✳ ❞❡ ▼❡❧♦ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ✲ ❙✉♣♣♦rt❡❞ ❜② ❋❆P✲❉❋ ●r♦✉♣s ❙t ❆♥❞r❡✇s ✷✵✶✼

  2. ❚❤✐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 C G ( ϕ ) t❤❡ ✜①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣ ♦❢ ϕ ✐♥ G ✱ C G ( ϕ ) = { x ∈ G ; x ϕ = x } .

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

  4. ❋✐①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣ ❉❡✜♥✐t✐♦♥ ◮ ▲❡t ϕ ❜❡ ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❛ ❣r♦✉♣ G ✳ ❲❡ ❞❡♥♦t❡ ❜② C G ( ϕ ) t❤❡ ✜①❡❞✲♣♦✐♥t s✉❜❣r♦✉♣ ♦❢ ϕ ✐♥ G ✱ C G ( ϕ ) = { x ∈ G ; x ϕ = x } . ❚❤✐s s✉❜❣r♦✉♣ ✐s ❛❧s♦ ❝❛❧❧❡❞ ✏t❤❡ ❝❡♥tr❛❧✐③❡r ♦❢ ϕ ✐♥ G ✑✳ ■❢ C G ( ϕ ) = ✶✱ ✇❡ s❛② t❤❛t ϕ ✐s ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ G ✳ ◮ ❙✐♠✐❧❛r❧②✱ ✐❢ A ≤ Aut ( G ) ✇❡ ❞❡♥♦t❡ ❜② C G ( A ) t❤❡ s✉❜❣r♦✉♣ C G ( A ) = { x ∈ G ; x a = x ❢♦r ❛❧❧ a ∈ A } . ■❢ C G ( A ) = ✶✱ ✇❡ s❛② t❤❛t A ✐s ❛ ✜①❡❞✲♣♦✐♥t✲❢r❡❡ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ G ✳

  5. ■❢ ✐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✳

  6. ■❢ ✐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❤❡♥ C G / N ( A ) = C G ( A ) N / N ❢♦r ❛♥② A ✲✐♥✈❛r✐❛♥t ♥♦r♠❛❧ s✉❜❣r♦✉♣ N ♦❢ G ✳

  7. ❙♦♠❡ ✉s❡❢✉❧ r❡s✉❧ts ✐♥ ❝♦♣r✐♠❡ ❛❝t✐♦♥s✳ ◮ ■❢ A ✐s ❛ ❣r♦✉♣ ♦❢ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ G ❛♥❞ ( | A | , | G | ) = ✶✱ t❤❡♥ C G / N ( A ) = C G ( 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 = � C G ( a ) ; a ∈ A # � ✳

  8. ✭ ❏✳ ❚❤♦♠♣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 ✲❜♦✉♥❞❡❞✳

  9. ✭ ❆✳ ❚✉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✳

  10. ✭ ❆✳ ❚✉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 C G ( ϕ ) ✐s ♥✐❧♣♦t❡♥t✱ t❤❡♥ ✐ts ❋✐tt✐♥❣ ❤❡✐❣❤t ✐s ❛t ♠♦st ✸✳

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