Let where is a finite group.

If , what conditions we need in order to have ???

conjugacy class?? order??

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- Feb 17th 2014, 06:27 AMdeniselim17if ab=ca, how to get b=c???
Let where is a finite group.

If , what conditions we need in order to have ???

conjugacy class?? order?? - Feb 17th 2014, 05:06 PMchiroRe: if ab=ca, how to get b=c???
Hey deniselim17.

Think about applying various post and pre multiplications of b inverse and c inverse. For example if we apply b inverse post multiplication on LHS we get a = cab^(-1) which is also equal to c^(-1)ab.

You can also get the identity in terms of a,b,c and b,c inverses.

This should give you a starting point to answering your question (you can use these identities to infer properties of a given that b = c). - Feb 18th 2014, 06:29 AMHartlwRe: if ab=ca, how to get b=c???
Given ab=ca

if b=c, ba=ca=ab

The condition for b=c is that G be commutative - Feb 18th 2014, 10:02 AMebainesRe: if ab=ca, how to get b=c???
- Feb 18th 2014, 01:46 PMSlipEternalRe: if ab=ca, how to get b=c???
Since the OP is dealing with groups, every element has an inverse. Hence, 0 is not an element of any multiplicative group, and there is no need to ensure . Hartlw is correct that the condition deals with commutativity. But, it is not necessary that the entire group be commutative. Instead, it is only necessary that where is the center of the group.

- Feb 18th 2014, 01:55 PMromsekRe: if ab=ca, how to get b=c???
- Feb 19th 2014, 05:41 AMHartlwRe: if ab=ca, how to get b=c???
- Feb 19th 2014, 06:07 AMSlipEternalRe: if ab=ca, how to get b=c???
- Feb 19th 2014, 06:17 AMDevenoRe: if ab=ca, how to get b=c???
So far, it appears that nobody has this right. While it is sufficient that G is abelian, or that a is in Z(G), it is not necessary. All that is required is that a and b commute, or (equivalently) a and c commute, nothing more.

To see a counter-example, let G = D_{4}, with a = r^{3}, b = c = r. None of these elements are in the center. - Feb 19th 2014, 06:27 AMSlipEternalRe: if ab=ca, how to get b=c???
- Feb 19th 2014, 07:10 AMHartlwRe: if ab=ca, how to get b=c???
- Feb 19th 2014, 07:18 AMSlipEternalRe: if ab=ca, how to get b=c???
As Deveno said, the only condition necessary is that $ab = ba$ (and equivalently $ca = ac$). Obviously, if $G$ is commutative, then $ab = ba$ for all $a,b \in G$. However, there are many non-commutative groups where there exists elements that commute. Both Deveno and I gave examples of such groups. So, while commutativity of $G$ is a

**sufficient**condition, it is not necessary. The only necessary condition is that the**specific**elements $a$ and $b$ commute. - Feb 19th 2014, 07:41 AMHartlwRe: if ab=ca, how to get b=c???
I didn't prove sufficiency, which is obvious but irrelevant to OP. I proved necessity, which answers OP.

- Feb 19th 2014, 07:54 AMSlipEternalRe: if ab=ca, how to get b=c???
- Feb 19th 2014, 08:07 AMHartlwRe: if ab=ca, how to get b=c???