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Handout Monday 10/27
Today we will continue to explore cosets and Lagrange’s Theorem. We should practice proving something is true about cosets. But then also, we will consider what a quotient group.
First, a quick example of Lagrange’s theorem. Suppose a subgroup of \(S_4\) contains all the 2-cycles and at least one 3-cycle. Must this be the entire group? Yes, because there are 6 2-cycles, and then also 4 elements that are the product of two disjoint 2-cycles. Any 3-cycle will get you it and its inverse. Plus the identity makes 13 elements. But the size of any subgroup must divide the size of the group, and \(S_4\) has size 24.
Another: Any group that has an element of order 3 and one of order 5 must have order a multiple of 15. This is because the element of order 3 generates a subgroup of size 3. The element of order 5 must generate a subgroup of size 5. The order of the group must be divisible by both 3 and 5, so must be divisible by 15.
Now work on the activity.
