So...an abelian group is both associative (because it's a group) and commutative (because it's abelian), which is exactly what the OP said? It sounds like you're disagreeing about something, but I'm not clear what your objection is.

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seanhunter 11 hours ago | parent [-]

I’m not disagreeing. I’m pointing out that in TFA it sounds as associativity is a property of abelian groups specifically whereas it as a property of all groups in general. In that sense it’s not wrong, just the emphasis is a bit misleading.

If you look in an abstract algebra textbook they all basically say the same definition for abelian groups (eg in Hien)

> “A group G is called abelian if its operation is commutative ie for all g, h in G, we have gh = hg”.

	▲	MarkusQ 6 hours ago \| parent [-]
		In an abstract algebra textbook, they define groups first and then abelian as a property that some groups have. Here, the author is defining abelian groups "from scratch" and doesn't have an earlier definition of groups to lean on. In more advanced texts, they could simply say that a group is a moniod with inverses and could (by your reasoning, should) avoid specifying that groups are associative since this is a property of all monoids.