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

anna010101

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.

Hurkyl

Well, what else could that exponent be?

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)

Just start writing down subgroups. I don't know what else to say.

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)

Count. If you don't know what to count, then count everything you can imagine, in as many different ways as you can imagine.

mathwonk

check my website.

or those of james milne, robert ash, ruslan sharipov, or just google your desired topic.