1. lagrange problem set

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

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

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

4. plz help

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

For example, the complete list of subgroups of $\mathbb{Z}_{24}$ 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?