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Unformatted text preview: S UMMARY OF 9/15 L ECTURE We started with a definition (motivated by examples). Let G be a group. We say H is a subgroup of G (denoted H G ) if: 1. H G ; and 2. H is a group under the same binary operation as G . Thus, although { 1 , 1 } is a subset of Z , and is a group under multiplication, ( { 1 , 1 } , ) is not a subgroup of ( Z , +) . On the other hand, it is a subgroup of ( Q , ) . Do you remember some of the other examples of subgroups we discussed? Yes or now, its a good exercise to look through our examples of groups from the first two lectures and try to come up with subgroups of them. Given a group G and a sub set S G , we defined the group generated by S to be the smallest subgroup of G containing S . (Smallest means: every group containing S must also contain the group generated by S .) In other words, given S , what other elements of G would you have to add to S to make it into a group (under the same binary operation as G )? This can be a somewhat subtle question in general, so we looked at a special case: what is the group generated by a single element x G ? Lets denote the binary operation of G by @ . We have to find the smallest subgroup of G containing x . Well, first of all, to be a group it has to contain the identity e of G . Also, to be a group it must have inverses, so x 1 has to be in there. What else? Well, any group is closed (i.e. satisfies the closure axiom: the combination of any two elements of the group lands somewhere in the group). So x @ x has to be in there. Similarly, x 1 @ x 1 has to be in there. What else? x @ x @ x and x 1 @ x 1 @ x 1 must be there, etc. In general, all elements of the form x @ x @ @ x  {z } n and x 1 @ x 1 @ @ x 1  {z } n have to be in there, for every positive integer n . Because this notation is getting unwieldy, we came up with a new notation: let x n = x @ x @ @ x  {z } n and x n = x 1 @ x 1 @ @ x 1  {z } n for every positive integer n . It will also be convenient to write x = e , the identity....
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This note was uploaded on 03/26/2011 for the course MAT 301 taught by Professor Gideonmaschler during the Fall '10 term at University of Toronto Toronto.
 Fall '10
 GideonMaschler

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