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.
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 isnít, and it has more elements than a (am≠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.