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**tttcomrader** For any element a in G, a group, prove that <a> is a subgroup of C(a).

Proof:

Now $\displaystyle <a> =$ {$\displaystyle e, a, a^2, ... , a^{n-1}$} if $\displaystyle |a| = n.$ So e would be in <a> for the least possible n, implies that <a> is nonempty. If <a> = {e}, then <a> is already a subgroup of C(a), so I assume <a> do not equal to {e}.

Let $\displaystyle a^i$ and $\displaystyle a^j$ be in <a>, and consider $\displaystyle C(a) = \{g \in G : ga = ag\ \forall a\}$

Now $\displaystyle (a^i)(a^j) = a^{i+j}$ , which is in <a>, and $\displaystyle (a^i)^{-1} = a^{-i}$, which is in <a> as well.

Furthermore, $\displaystyle (a^i)(a) = a^{i+1} = a^{1+i} = (a)(a^{i})$, so $\displaystyle a^{i} \in C(a)$ , thus proves <a> is a subgroup of C(a).

Q.E.D.

Is that right?