# Thread: lagrange problem set

1. ## lagrange problem set

2. For #1, let $\displaystyle N_1,N_2$ be normal subgroups of orders $\displaystyle 3\mbox{ and }5$ respectively. Using the identity $\displaystyle |N_1N_2||N_1\cap N_2| = |N_1||N_2|$ and the fact that $\displaystyle N_1\cap N_2 = \{ e\}$ we find that $\displaystyle N_1N_2$ is a subgroup of order $\displaystyle 15$. Let $\displaystyle N_1N_2 = H$ then $\displaystyle N_1,N_2\triangleleft H$ with $\displaystyle N_1\cap N_2 = \{ e\}$ this means $\displaystyle H\simeq N_1\times N_2 \simeq \mathbb{Z}_3\times \mathbb{Z}_5\simeq \mathbb{Z}_15$. Therefore this group is cyclic of order $\displaystyle 15$.

Note, $\displaystyle D_5$ has ten elements but it is not cyclic.

3. For #2, let $\displaystyle G$ be a group of prime order. Let $\displaystyle a\in G$ so that $\displaystyle a\not = e$. Argue that $\displaystyle \left< a \right> = G$ using Lagrange's theorem.

4. ## plz help

please help me with second question last part....

5. Originally Posted by szpengchao
please help me with second question last part....
The general result says that the subgroups of $\displaystyle \mathbb{Z}_n$ (the cyclic subgroup of order $\displaystyle n$) are precisely $\displaystyle k\mathbb{Z}_n = \left< [k]_n\right>$ where $\displaystyle k$ is a positive divisor of $\displaystyle n$. And furthermore, $\displaystyle k\mathbb{Z}_n$ is a subgroup of $\displaystyle m\mathbb{Z}_n$ if and only if $\displaystyle m|k$.

For example, the complete list of subgroups of $\displaystyle \mathbb{Z}_{24}$ are given below.

6. ## help

what is the triangle in question 1, N_1 , N_2, triangle H

is that : belongs to?

7. Originally Posted by szpengchao
what is the triangle in question 1, N_1 , N_2, triangle H

is that : belongs to?
It means normal subgroup.

8. ## help

what does [k]_n means?