Prove that g has a non-trivial centre

Reference no: EM13193843

Let G be a group of order p^k(p prime, k > 0)

(a)Prove that G has a non-trivial centre (hint: use the class equation)

(b)Prove that G has a normal subgroup of order p^m for all m < k

Hint: You may use the following intermediate steps:
1. Prove that for any group G, any subgroup of the centre Z(G) is normal in G
2. Use the above and the rst part to nd a normal subgroup H of order p
3. Use induction on k to nd normal subgroups of G=H
4. Prove that for any group homomorphism p : G ! H and any normal subgroup K of H, p(-1)(K)is normal in G

(c)Prove that any group of order p^2 is Abelian

Reference no: EM13193843

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