Abstract Algebra: Theory and Applications · Problem 15.4.25

Rings — Thomas Judson

Mathematics — Abstract Algebra · sylow-theorems

Problem

Another Proof of the Sylow Theorems. (a) Suppose

p

is prime and

p

does not divide

m

. Show that

p \nmid \binom{p^k m}{p^k}.

(b) Let

\mathcal{S}

denote the set of all

p^k

element subsets of

G

. Show that

p

does not divide

|\mathcal{S}|

. (c) Define an action of

G

\mathcal{S}

by left multiplication,

aT = \{at : t \in T\}

for

a \in G

and

T \in \mathcal{S}

. Prove that this is a group action. (d) Prove

p \nmid |\mathcal{O}_T|

for some

T \in \mathcal{S}

. (e) Let

\{T_1, \ldots, T_u\}

be an orbit such that

p \nmid u

and

H = \{g \in G : gT_1 = T_1\}

. Prove that

H

is a subgroup of

G

and show that

|G| = u|H|

. (f) Show that

p^k

divides

|H|

and

p^k \leq |H|

. (g) Show that

|H| = |\mathcal{O}_T| \leq p^k

; conclude that therefore

p^k = |H|