# Group definition does not make sense to a group theorist

From Louis de Focrand:

I don’t know if this is the right place for this, but I just read through the beginning of a few sections, and looking at the one on groups I found the mathematical definition of a group kind of peculiar, in particular point (c):

« every element e has a left and right inverse, that is there are elements e' and e with the properties that ee'=ee=I. If left and right inverses are equal for all a the group is symmetric and is known as an Abelian group. »

While different sources surely have their own conventions, both my mathematics course on abstract algebra as well as Wikipedia agree that the left and right inverses must be equal, and (more importantly) that an Abelian group G is a commutative group, that is that ab = ba for all a and b in G. See https://en.m.wikipedia.org/wiki/Abelian_group#Definition

From Ewart Shaw:

Note that the left and right inverses of a group are automatically equal [consider e"ee': e" = e"I = e"(ee') = (e"e)e' = Ie' = e']. The definition of Abelian in Chapter 26 is incorrect: an Abelian group G is one for which ab = ba for all elements a, b in G.

I'd also suggest using, say, a rather then e for a single generic element of the group, as e is often used to denote the identity.

Noted.

Moreover, the word "symmetric" is confusing in this context, inviting confusion with a particular kind of finite group called the Symmetric Group on n points: Sym(n).

I propose to rephrase (c) with reference to an authoritative definition of a group. E.g. http://mathworld.wolfram.com/Group.html

Ian Clark (talk) 19:08, 28 May 2018 (UTC)