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LS9-15 - S UMMARY OF 9/15 L ECTURE We started with a...

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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, it’s 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 ? Let’s 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 0 = e , the identity.
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