- #1
Firepanda
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Z(G) = { x in G : xg=gx for all g in G } (center of a group G)
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 x1, x2,... in G when composed with every element in g, commute.
The second set tells me what x1, x2,... 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_subgroupSo how does this differ from C(g), what other steps do I need?
Thanks
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 x1, x2,... in G when composed with every element in g, commute.
The second set tells me what x1, x2,... 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_subgroupSo how does this differ from C(g), what other steps do I need?
Thanks