Group homomorphisms erratum
Okay, it’s been pointed out to me that what I was thinking of in my update to yesterday’s post was a little more general than group theory. In the case of groups, preserving the composition is all that’s required. If I talk about semigroups later that extra condition will be needed.
So, I’ll leave it as a (relatively straightforward) exercise to show that if a function from one group to another preserves the composition that it also preserves identities. Oh, and inverses. May as well nail that down while we’re at it.
For now I’ll make this point, which I also make to all my calculus classes: it’s really not that bad to misremember something. That’s one of the nice things about mathematics. If you make a mistake it’s usually not hard to check. Also, when two people have different views on something it’s possible to check which one is right. There are real answers to be found. Despite the accusations of coldness and impersonality, it’s comforting to know that something has a real answer.
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