# iven any group G and a subset W, let < W > be the smallest subgroup of G which contai

• Mar 17th 2010, 07:04 PM
snick
iven any group G and a subset W, let < W > be the smallest subgroup of G which contai
Given any group G and a subset W, let < W > be the smallest subgroup of G which contains W?

A) prove that there is such a subgroup < W > in G. ( < W > is called the subgroup generated by W.)

B) If gwg ^ -1 € W for all g € G, w € W, prove that < W > is a normal subgroup of G.

C) Now, let U={xyx^ -1y^-1 | x, y € G}. Prove that < U > is normal in G.
• Mar 17th 2010, 07:10 PM
Drexel28
Quote:

Originally Posted by snick
Given any group G and a subset W, let < W > be the smallest subgroup of G which contains W?

A) prove that there is such a subgroup < W > in G. ( < W > is called the subgroup generated by W.)

B) If gwg ^ -1 € W for all g € G, w € W, prove that < W > is a normal subgroup of G.

C) Now, let U={xyx^ -1y^-1 | x, y € G}. Prove that < U > is normal in G.

A) Let $\mathcal{M}=\left\{H:W\subseteq H\leqslant G\right\}$. Define $K=\bigcap_{M\in\mathcal{M}}M$

B) What definition do you use?

C) What have you tried?