1. For each g in G, show that y is an element of xZ(g) (left coset of the centralizer) if and only if ygy^{-1}= xgx^{-1}

2. Show that y is an element of xZ(G) (left coset of the center) if and only if ygy^{-1}= xgx^{-1}for all g in G.

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- Oct 24th 2012, 09:27 PMTheHowlingLungCosets
1. For each g in G, show that y is an element of xZ(g) (left coset of the centralizer) if and only if ygy

^{-1}= xgx^{-1}

2. Show that y is an element of xZ(G) (left coset of the center) if and only if ygy^{-1}= xgx^{-1}for all g in G. - Oct 24th 2012, 09:46 PMjohnsomeoneRe: Cosets
1) Do you know the definition of the centralizer of g?

2) Do you know the definition of the center of a group?

3) Do you know the definition of a coset?

These two problems are just straightforward consequences of the definitions.

Maybe this will help: a is in bH (where H is a subgroup of some group G, a and b are elements of G) if and only if (b-inverse)(a) is in H. - Oct 25th 2012, 03:47 AMTheHowlingLungRe: Cosets
I know the definitions, I'm just having trouble seeing how to use them to show that those statements are true.

The centralizer of g in G is the set of elements of G which commute with g. The center of a group is the set of elements that commute with every element of G. A coset is basically a shifted subgroup of G, more specifically gH = {gh: h E G} where H is a subgroup of G.

I know how to find centralizers, centers, and cosets of a group, but I'm not really sure how to show these proofs. - Oct 25th 2012, 08:12 AMDevenoRe: Cosets
this can be rephrased as follows:

different conjugates of g give rise to distinct cosets of Z(g). think about what this may mean. anyway:

suppose y is in xZ(g). what does this mean? it means x^{-1}y is in Z(g). so x^{-1}y commutes with g:

x^{-1}yg = gx^{-1}y

x^{-1}ygy^{-1}= gx^{-1}(multiplying on the right by y^{-1})

ygy^{-1}= xgx^{-1}(multiplying on the left by x).

that's what we wanted to prove for y in xZ(g) implies ygy^{-1}= xgx^{-1}.

the other implication merely runs this argument backwards.

that's part (1) of your problem.

part (2) is an easy generalization. for if x^{-1}y commutes with every element g of G, it surely lies in Z(G).