C. Prove for any two elements x,y∈G that xy=yx.

Asked Oct 13 '16 at 01:40

Active Oct 13 '16 at 01:40

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Suppose the quotient group $G/C$ is cyclic and generated by element $Ca$ of $G/C$. Prove for any two elements $x,y \in G$ that $xy=yx$.

I am having trouble formulating a proof. In a preceding problem I was able to prove that for every $x \in G$ there is some integer $m$ such that $x=ca^m$ which I am guessing I can use to write $y=c'a^n$. Where would I go from here?

asked Oct 13 '16 at 01:40

Ddbb1994

Is $C$ anything special? If $C$ is the center, then $cg = gc$ and $c'g = gc'$ for all $g \in G$. – J126 Oct 13 '16 at 01:43
Ahh sorry I forgot to add that! C is the center of the group G. – Ddbb1994 Oct 13 '16 at 01:47
So multiply $xy=(ca^m)(c'a^n)$ that seems a natural way to start. Then use properties of the center. – Crenner Oct 13 '16 at 02:15

Suppose the quotient group G/C is cyclic and generated by element Ca of G/C. Prove for any two elements x,y∈G that xy=yx.

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