# Groups elements

Let $G=S_3 \times \mathbb{Z}_4$ be a group and $H= \langle(1 2 3)\rangle \times \langle2\rangle$ and $K=\langle(1)\rangle \times \langle2\rangle$ be subgroups.

1. Show in two different ways that H normal to G, k is normal to H and K normal to G

2. Find the elements of H/K and G/H

3. Find an isomorphic group to G/H, and prove it with the isomorphism theorem

• thanks for your help....... if im asking is because i dooooon't understand it !!!!!! and i really need it for tomorrow!!! Dec 5 '13 at 3:24
• Of course, but you are expected to at least show what it is you don't understand, where are you stuck, what you've tried to solve the problem, &c. Dec 5 '13 at 3:37
• first I dont understand my group G I Know that S3= (1)(12)(13)(23)(123)(321) order 6 and Z4=0123 but s3 xZ4 ?? Dec 5 '13 at 3:43
• In your previous question you were explained a list of things. Did you read that carefully? Dec 5 '13 at 3:44
• I did!... this is for take home test for tomorros and i have my final exam friday!! im trying to understand it!! Dec 5 '13 at 3:48

1. To see that $H \triangleleft G$, let $(\sigma, n) \in G$. Show that $(\sigma, n) H = H (\sigma, n)$. By property of direct products, it suffices to show separately that both $\sigma \langle (123) \rangle = \langle (123) \rangle \sigma$ and $n+2 = 2+n$. A similar process nets you normality of the other two.
2. Elements of $H/K$ are of the form $((1), 0 \bmod 2)$ and elements of $G/H$ are of the form $(\tau \bmod (123), n \bmod 2)$.
3. Hint: in my definition of $G/H$, $\tau$ is a transposition.
• $(\tau \circ \langle(123) \rangle, n \bmod 2)$. Dec 5 '13 at 4:38
• Hmm, since $\langle (123) \rangle$ is cyclic, that notation really should be $(\tau \bmod (123), n \bmod 2)$. I've edited my response. Dec 5 '13 at 4:51