The OP's original question began with "Let $a,b,c \in G$". That seems like the OP has a specific $a$, and is not looking for a condition where it will be true for all $a$. Perhaps the OP could clarify?

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- February 19th 2014, 08:13 AMSlipEternalRe: if ab=ca, how to get b=c???
- February 19th 2014, 08:28 AMHartlwRe: if ab=ca, how to get b=c???
Why did you wait till now to ask for a clarification of OP? It seems to have been accepted by everyone up till now.

Anyhow, have to leave for a long trip to my mother-in-law. Please do not interpret my silence as acquiesence to all the unfounded bloated obtuse objections. - February 21st 2014, 07:40 AMHartlwRe: if ab=ca, how to get b=c???
The existence of an identity element does not contradict the above, as proposed in a disguised and obscured previous misguided post. Translated, the post says eb=ce →b=c without requiring G be commutative, which as an argument against commutatvity is of course nonsense. ae=ea is part of definition of a group. To add insult to injury, it was then suggested we ask OP how many elements in the group.

Another obfuscation, G generated by a, is irrelevant. G is either a group or it isn’t, and it has more elements than a (a^{m}≠a).

I used the definition of group to answer the OP, which seems to have generated a lot of confusion and/or nastiness. Perhaps the confused should look up the definition of a group.

There seems to be a conception among some that irrelevant and unnecessary abstraction is a sign of honesty and intelligence- on the contrary.