## Lagrange's Theorem (Order of a group) Abstract Algebra

Can someones tells me how to prove these theorems.
1. Prove that if G is a group of order p^2 (p is a prime) and G is not cyclic, then a^p = e (identity element) for each a E(belongs to) G.

2. Prove that if H is a subgroup of G, [G:H]=2, a, b E G, a not E H and b not E H, then ab E H.

3. Verify that S4 has at least one subgroup of order k for ech divisor of 24

4. If H is a subgroup of G and [G:H] = 2, the the right cosets of H in G are the same as the left cosets of H in G. Why?

5. Prove that if H is a subgroup of a finite grup G, the n the number of right cosets of H in G equals the number of left cosets of H in G.

Also, does anyone know a good website that has good information for abstract algebra. Thanks.
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 1. Prove ... a^p = e
Well, what else could that exponent be?

 2. Prove that if H is a subgroup of G, [G:H]=2, a, b E G, a not E H and b not E H, then ab E H.
If you're only worried about whether or not something is in H... then why don't you work with G/H instead? (You know H is normal, right? If not, then see #4)

 3. Verify that S4 has at least one subgroup of order k for ech divisor of 24
Just start writing down subgroups. I don't know what else to say.

 4. If H is a subgroup of G and [G:H] = 2, the the right cosets of H in G are the same as the left cosets of H in G. Why?
You should know a very simple description of the left and right cosets of H. (Start by counting them -- you know how many there are)

 5. Prove that if H is a subgroup of a finite grup G, the n the number of right cosets of H in G equals the number of left cosets of H in G.
Count. If you don't know what to count, then count everything you can imagine, in as many different ways as you can imagine.
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