Recent content by tyrannosaurus

Homework Statement Assume that G is a group of order 63 that has two Sylow subgroups whose intersecion is non-trivial. Show that G has an element of order 21. Homework Equations The Attempt at a Solution So by Sylow's theroem, I know that 63=3^2*7, that the sylow 7 subgroup is...

How about this: since x^n=y^2, then G= <x> union y<x>. So G= {y^jx^i|0<=i<= n-1, 0<=j<=1}. Is x^k an element of Z(G) for some k? So x^ky=yx^k, thus k=2 since x^2=y (do I need to say more?). Thus x^k is contained in the Z(G). Is yx^k an element of Z(G) for some K? The answer is no, but how...

Homework Statement Let G=<x, y| x^{2n}=e, x^n=y^2, xy=yx^{-1}>. Show Z(G)={e, x^n}. Homework Equations The Attempt at a Solution So I tried breaking this up into cases: Case 1: If n=1. then |x|=1 or 2. If |x|=1, then x=e and x would obviously be in the center. If |x|=2, then xy=yx (since...

Thanks for the help, I just needed some clarification.

Homework Statement Let H and K be normal Subgroups of a group G s.t H intersect K = {e}. Show that G is isomorphic to a subgroup of G/H + G/K. Homework Equations G/H+G/K= direct product of G/H and G/K. The Attempt at a Solution Proof/ Lets define are mapping f:G to G/H+G/K by...

thanks for your help, that helped a lot.

Homework Statement Let G=<a, b|a^2=b^2=e, aba=bab>. SHow G is isomorphic to S3. Homework Equations [b]3. The Attempt at a Solution [/ since a^2=b^2=e, then |a|=1 or 2 and |b|= 1 or 2. But since aba=bab, the orders of a and b both have to be 2 because if either had order 1, we would...

Homework Statement What is the minimum number of generators needed for Z2+Z2+Z2? Find a set of generators and relations for this group. Homework Equations The Attempt at a Solution I think it is obvious that the minimum amount of generators that you need is three, with Z2+Z2+Z2 =...

Z(G) is the center of the group G, for a to be an element of Z(G) a has to commute with all elements of G. Z(G) can be trivial (i.e just e) in some cases, but in this one it can't since |G|=p^n where p is a prime (p0groups have non-trivial centers).

{e} is always a subgroup, but in are case Z(G) cannot equal e since are group G is of order p^n, where p is a prime.

1, p, p to m where m divides n and p^n. We know that we can't have 1 because Z(G) needs more than one lement

Homework Statement Suppose the G is a group of order p^n, where p is prime, and G has exactly one subgroup for each divisor of p^n. Show that G is cyclic. Homework Equations The Attempt at a Solution First, the inductive basis holds for n=1. Now for each k>=1, we assume that the...

A general fact is that |KS| = |K| |S| / |K intersect S|. I forgot about this fact! THanks so much, this makes a whole lot of more sense now.

Homework Statement Suppose K is a normal subgroup of a finite group G and S is a p-Sylow subgroup of G. Prove that K intersect S is a p-Sylow subgroup of K. So I know that K is a unique p-sylow group by definition, is that enough to prove that the intersection of K with S is a p-sylow...

Homework Statement Show that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28. Homework Equations The Attempt at a Solution Let H be a normal subgroup of order 4. Then |G/H|=42=2*3*7, so then G?N has a unique, and...