Z(G) = { x in G : xg=gx for all g in G } (center of a group G)(adsbygoogle = window.adsbygoogle || []).push({});

C(g) = { x in G : xg=gx } (centralizer of g in G)

I have to show both are subgroups, but what's the difference in the methods?

To me the first set is saying all the elements x_{1}, x_{2},... in G when composed with every element in g, commute.

The second set tells me what x_{1}, x_{2},... in G when composed with a single chosen element from g in G, commute.

Is this correct?

I've found the way to show Z(G) is a subgroup from here :

http://en.wikipedia.org/wiki/Center_(group_theory)#As_a_subgroup

So how does this differ from C(g), what other steps do I need?

Thanks

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# Homework Help: I can't tell the difference between these groups :/

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