Let G be a group of order 45. Explain why the centre of G cannot have order 9 or 15

Asked Dec 05 '24 at 14:08

Active Dec 05 '24 at 14:08

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I am not sure how to I tried to use the class equation to prove that the centre $Z(G)$ of a group $ G$ of order 45 cannot have order 9 or 15, but I’m struggling with how to proceed, especially since ( G ) is not specified as non-abelian.

I tried to use the class equation: $$ |G| = |Z(G)| + \sum_{i} [G : C_G(x_i)], $$ where the sum is over representatives $ x_i $ of the conjugacy classes of $ G $ that are not in the centre, and $ C_G(x_i) $ is the centralizer of $ x_i $.

I tried using this to prove that $|Z(G)|$ couldn't be 20, but I am not sure how to proceed.

Any help would be greatly appreciated.

asked Dec 05 '24 at 14:08

Jake O Grady

1

You asked the same question here just a few minutes ago. Why are you asking it here again? – Soumik Mukherjee Dec 05 '24 at 14:12
I am trying to put up a better post, that explains what I have worked on better – Jake O Grady Dec 05 '24 at 14:14
Do you know that if G/Z(G) is cyclic then G is abelian? – Soumik Mukherjee Dec 05 '24 at 14:15
I get that now thank you – Jake O Grady Dec 05 '24 at 14:18
Is the class equation above not useful at all in this type of question? – Jake O Grady Dec 05 '24 at 14:24
Please do not delete then repost; instead, edit the existing question. – Shaun Dec 05 '24 at 15:14
The group is abelian. See here: https://math.stackexchange.com/q/153657/104041 – Shaun Dec 05 '24 at 15:24
Clearly the OP wished to pursue a different approach than the one in the dupe target. – Kan't Dec 05 '24 at 18:40
If the center has order $9$ or $15$ then there is no room for the centralizers of the noncentral elements (Lagrange). – Kan't Dec 05 '24 at 18:45
If fact, for $g\notin Z(G)$, $Z(G)<C_G(g)<G$. But if $|Z(G)|=9$ there is no proper multiple of $9$ which is also a proper divisor of $45$. And the same if $|Z(G)|=15$. – Kan't Dec 05 '24 at 20:02

Let G be a group of order 45. Explain why the centre of G cannot have order 9 or 15

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