SL(n, R) in GL(n, R), ((1 1 0 1)) in GL( 2,Z_2). Let G be a group and let H be a
ID: 1719750 • Letter: S
Question
SL(n, R) in GL(n, R), ((1 1 0 1)) in GL( 2,Z_2). Let G be a group and let H be a subgroup. Show |G/H| = |HG| (here HG stands for the set of right cosets). For n epsilon N, show there exits a subgroup H of (Q^+, ) so that |Q^+/H| = n. Suppose H Q is a subgroup of Q. Show |Q/H| = infinity. Now look at n q/n. Conclude that Q^+ Q. What are the possible orders of subgroups of: Z_20, D_6, S_5, Z_12 Times S_3. Let p epsilon N be a prime and let G be a group with |G| = p. Show G has exactly two subgroups: {e} and G with nothing in-between. If H and K are finite subgroups of G, show |H K| divides (|H|, |K|). If (|H|, |K|) = 1, show H K = {e}. If |H| is prime, show H K is either {e} or H. Suppose K H G is a chain of subgroups. Show |G/K| = IG/H| |H/k|. Let G be a finite abelian group and let p be a positive prime dividing the order of G. ThisExplanation / Answer
Given that order of G is p
Since p is prime p has only two factors p and 1
By theorem on subgroups, we have order of a subgroup divides the order of the group
Hence subgroups of G can have orders only as 1 and p
If order is 1, then as identity element has to be in subgroup, the subgroup can have only {e} as a group
The next one is order p , hence the subgroup is the entire group G
No other subgroup is possible for G
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