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math.ucr.edu · 3,059 words · saved by 1 readers

This definition is fairly mysterious at first. It becomes a bit less mysterious when you learn that N is normal in G precisely when you can "mod out" G by N and get a new group G/N. Guys in G/N are equivalence classes of guys in G. Two guys in G, say g and g', define the same guy in G/N precisely when g' = gn for some n in N. The condition that N be normal is precisely what we need to get a well-defined way of multiplying these equivalence classes! But what are normal subgroups like? How can you just look at a subgroup and guess if it's normal? Most people don't have the fortitude to think about this question very hard! Shawn Fitzpatrick did, and he sent me an email about it: Hi there. My name's Shawn Fitzpatrick. I'm currently in the process of trying to come to grips with a few modern algebra concepts, and I came across your website, http://math.ucr.edu/home/baez/week155.html. From the website, I got the impression that you're a kindred spirit, in so far as you seem to crave a visual

normal What's a Normal Subgroup? John Baez February 8, 2005 If you've studied group theory, you'll probably know that a subgroup N of a group G is called "normal" if conjugating any element of N by an element of G gives another element of N. In other words, if n is in N and g is in G, then gng -1 is in N. This definition is fairly mysterious at first. It becomes a bit less mysterious when you learn that N is normal in G precisely when you can "mod out" G by N and get a new group G/N. Guys in G/N are equivalence classes of guys in G. Two guys in G, say g and g', define the same guy in G/N preci

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