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Let $G=K_1\times K_2\times K_3\times K_4\times ......\times K_n$ and $g\in G$ Then I have to describe Center and $N(g)$ in terms of $K_i$. Where $$N(g)=\{x\in G| gx=xg\}.$$ My attempt: $t\in Z(G)$ therefore $tm=mt$ for all $m \in G$ So $t=(t_1,t_2,......,t_n)$ and $m=(m_1,m_2,......,m_n)$ So individually componentwise we get $(t_im_i)=(m_it_i)$. So $(t_i)\in Z(K_i)$. Is this is sufficent?

Also on same account $s\in N(g)$ then $s_i \in N(g_i)$ Is my arguments are right or need some extra work? Any help will be appreciated.

Hanul Jeon
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Curious student
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  • You haven't made it very clear what your conclusion for center is - is it $Z(K_1)\times\cdots\times Z(K_n)$? The argument for this should be 'if and only if', so you need to make it clear that your logic is both ways. Otherwise the argument seems correct and the analogous argument does work for $N(g)$. – Robert Chamberlain May 06 '18 at 07:52
  • The part with the centre is a duplicate. – Dietrich Burde May 06 '18 at 08:03

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