Alhebra. Shdhhshshshsbsbsbsbq tmb hj hj hj já

❈❧❛ss✐✜❝❛çã♦ ❞❡ ❣r✉♣♦s s✐♠♣❧❡s ✜♥✐t♦s ❞❡ ♦r❞❡♠ ✐♥❢❡r✐♦r ❛ ✶✵✵
❈❧❛✉❞✐❛ ❈♦rr❡❛ ❞❡ ❆♥❞r❛❞❡ ❖❧✐✈❡✐r❛✶
P❛✉❧♦ ❆❣♦③③✐♥✐ ▼❛rt✐♥✭❖r✐❡♥t❛❞♦r✮✷

✸✳ ❙❡ ❑ é s✉❜❣r✉♣♦ ❞❡ ● ❝♦♠ | K |= pi (i ≤ n)✱
❡♥tã♦ ❡①✐st❡ H ∈ Sylp (G) t❛❧ q✉❡ ❑ é s✉❜❣r✉♣♦
❞❡ ❍✳

❡

✶ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ✭❯❙P✮✱ ❇r❛s✐❧

✹✳ ❙❡ np =| Sylp (G) |✱ ❡♥tã♦ np | m ❡ np ≡ 1 ✭♠♦❞
♣✮✳

❝❧❛✉❞✐❛❅❧✐♥✉①✳✐♠❡✳✉s♣✳❜r

✷ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ✭❯❙P✮✱ ❇r❛s✐❧

❛❣♦③③✐♥✐❅✐♠❡✳✉s♣✳❜r

Pr♦♣♦s✐çã♦ ✷✳✹✳ ❙❡❥❛ ● ✉♠ ❣r✉♣♦ t❛❧ q✉❡ | G |= p✱
❝♦♠ ♣ ♣r✐♠♦✳ ❊♥tã♦ ● é ❝í❝❧✐❝♦ ❡ s✐♠♣❧❡s✳

✶✳

Pr♦♣♦s✐çã♦ ✷✳✺✳ ❙❡❥❛♠ ● ✉♠ ❣r✉♣♦ ❡ ❍ s✉❜❣r✉♣♦

■♥tr♦❞✉çã♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♦ ❡st✉❞♦ ❞❡
❣r✉♣♦s s✐♠♣❧❡s ✜♥✐t♦s✳ ❊♥❢♦❝❛♠♦s ❛ss✉♥t♦s ❝♦♥❝❡r✲
♥❡♥t❡s à ❝❧❛ss✐✜❝❛çã♦ ❞❡ss❡s ❣r✉♣♦s✱ ✐st♦ é✱ r❡s✉❧t❛❞♦s
❡ té❝♥✐❝❛s q✉❡ ♥♦s ♣❡r♠✐t❛♠✱ ❛ ♣❛rt✐r ❞❛ ♦r❞❡♠ ❞❡
✉♠ ❣r✉♣♦ ❛❜str❛t♦✱ ❝❧❛ss✐✜❝á✲❧♦ ♦✉ ♥ã♦ ❝♦♠♦ s✐♠✲
♣❧❡s✳ ❚r❛❜❛❧❤❛♠♦s ❝♦♠ ❣r✉♣♦s ❞❡ ♦r❞❡♥s ✐♥❢❡r✐♦r❡s
❛ ✶✵✵✳
P❛r❛ ♦ ❡st✉❞♦ ❞❡ss❛s ♦r❞❡♥s✱ ❝♦♥str✉í♠♦s té❝♥✐❝❛s
❜❛s❡❛❞❛s ❡♠ r❡s✉❧t❛❞♦s ❝♦♠♦ ♦s ❚❡♦r❡♠❛ ❞❡ ❙②❧♦✇
❡ ❚❡♦r❡♠❛ ❞❡ ❈❛✉❝❤②✳
❈♦♠♦ ❡①❡♠♣❧♦ ❞❡ ❣r✉♣♦ s✐♠♣❧❡s✱ ❡st✉❞❛♠♦s ❛ ❢❛✲
♠í❧✐❛ An ✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♠♦str❛♠♦s q✉❡ t♦❞♦ ❣r✉♣♦ s✐♠♣❧❡s ❞❡ ♦r❞❡♠ ✻✵ é ✐s♦♠♦r❢♦ ❛♦ A5 ✳
✷✳

❞❡ ● t❛❧ q✉❡ [G : H] = 2✱ ❡♥tã♦ ❍ é ♥♦r♠❛❧ ❡♠ ●✳

Pr♦♣♦s✐çã♦ ✷✳✻✳ ❙❡❥❛ ● ✉♠ ❣r✉♣♦ t❛❧ q✉❡ | G |= p2 ✱
❝♦♠ ♣ ♣r✐♠♦✳ ❊♥tã♦ ● é ❛❜❡❧✐❛♥♦✳

Pr♦♣♦s✐çã♦ ✷✳✼✳ ❙✉♣♦♥❤❛ q✉❡

| G |= pm ✱ ❝♦♠ ♣

♣r✐♠♦ ❡ n ≥ 2✳ ❊♥tã♦ ● ♥ã♦ é s✐♠♣❧❡s✳

Pr♦♣♦s✐çã♦ ✷✳✽✳ ❙❡❥❛ ● ✉♠ ❣r✉♣♦ ✜♥✐t♦✳ ❙❡❥❛ ❑

s✉❜❣r✉♣♦ ❞❡ ● t❛❧ q✉❡ [G : H] = m✳
❙✉♣♦♥❤❛ q✉❡ | G | m!✱ ❡♥tã♦ ❡①✐st❡ H = {❡} ✭♦♥❞❡
❡ é ❛ ✐❞❡♥t✐❞❛❞❡ ❞♦ ❣r✉♣♦✮ t❛❧ q✉❡ ❍ é s✉❜❣r✉♣♦ ❞❡
❑ ❡ ❍ é ♥♦r♠❛❧ ❡♠ ●✳

Pr♦♣♦s✐çã♦ ✷✳✾✳ ❙❡❥❛ ● ✉♠ ❣r✉♣♦✳ ❙✉♣♦♥❤❛ q✉❡

| G |= 2.n✱ ♦♥❞❡ ♥ é í♠♣❛r✱ n ≥ 3✳ ❊♥tã♦ ● ♥ã♦ é

s✐♠♣❧❡s✳

Pr♦♣♦s✐çã♦ ✷✳✶✵✳ ❙❡❥❛ ● ✉♠ ❣r✉♣♦ t❛❧ q✉❡ | G |=

❘❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s

◆❡ss❛ s❡çã♦✱ ❡♥✉♥❝✐❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s
✉t✐❧✐③❛❞♦s ♥❛ ❝♦♥str✉çã♦ ❞❛s té❝♥✐❝❛s q✉❡ ❞❡s❡♥✈♦❧✲
✈❡♠♦s ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳ ◆ã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛s
❞❡♠♦♥str❛çõ❡s✱ ♣♦✐s sã♦ ❢❛❝✐❧♠❡♥t❡ ❡♥❝♦♥tr❛❞❛s ❡♠
❧✐✈r♦s ❜ás✐❝♦s ❞❡ á❧❣❡❜r❛✳

pq ✱ ♣