GH = {g e G|gHg- = H} is a subgroup of G. 12. Let G be a finite group and let primes p and q # p divide |G|. Prove that if G has precisely one proper Sylow p-subgroup, it is a normal subgroup, so G is not simple. 12 Show that every group of order 45 has a normal subhoroun of order 9.

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Section 36 number 12
that GH = {g e G|gHg = H} is a subgroup of G.
12. Let G be a finite group and let primes p and q p divide |G|. Prove that if G has precisely one proper Sylow
p-subgroup, it is a normal subgroup, so G is not simple.
13 Show that every group of order 45 has a normal subgroup of order 9.
4. Prove Corollary 364
EC Show thot
Transcribed Image Text:that GH = {g e G|gHg = H} is a subgroup of G. 12. Let G be a finite group and let primes p and q p divide |G|. Prove that if G has precisely one proper Sylow p-subgroup, it is a normal subgroup, so G is not simple. 13 Show that every group of order 45 has a normal subgroup of order 9. 4. Prove Corollary 364 EC Show thot
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