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# Section4.3Exercises

##### 1

True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let $G$ and $G'$ be groups.

1. Every group contains at least two distinct subgroups.

2. If $H$ is a proper subgroup of group $G$ and $G$ is finite, then we must have $|H|\lt |G|\text{.}$

3. $7\Z$ is a subgroup of $14\Z\text{.}$

4. A group $G$ may have two distinct proper subgroups which are isomorphic (to one another).

##### 2

Give specific, precise examples of the following groups $G$ with subgroups $H\text{:}$

1. A group $G$ with a proper subgroup $H$ of $G$ such that $|H|=|G|\text{.}$

2. A group $G$ of order 12 containing a subgroup $H$ with $|H|=3\text{.}$

3. A nonabelian group $G$ containing a nontrivial abelian subgroup $H\text{.}$

4. A finite subgroup $H$ of an infinite group $G\text{.}$

##### 3

Let $n\in \Z^+\text{.}$

1. Prove that $n\Z \leq \Z\text{.}$

2. Prove that the set $H=\{A\in \M_n(\R)\,:\,\det A=\pm 1\}$ is a subgroup of $GL(n,\R)\text{.}$

(Note: Your proofs do not need to be long to be correct!)

##### 4

Let $n\in \Z^+\text{.}$ For each group $G$ and subset $H\text{,}$ decide whether or not $H$ is a subgroup of $G\text{.}$ In the cases in which $H$ is not a subgroup of $G\text{,}$ provide a proof. (Note. Your proofs do not need to be long to be correct!)

1. $G=\R\text{,}$ $H=\Z$

2. $G=\Z_{15}\text{,}$ $H=\{0,5,10\}$

3. $G=\Z_{15}\text{,}$ $H=\{0,4,8,12\}$

4. $G=\C\text{,}$ $H=\R^*$

5. $G=\C^*\text{,}$ $H=\{1,i,-1,-i\}$

6. $G=\M_n(\R)\text{,}$ $H=GL(n,\R)$

7. $G=GL(n,\R)\text{,}$ $H=\{A\in \M_n(\R)\,:\,\det A = -1\}$

##### 5

Let $G$ and $G'$ be groups, let $\phi$ be a homomorphism from $G$ to $G'\text{,}$ and let $H$ be a subgroup of $G\text{.}$ Prove that $\phi(H)$ is a subgroup of $G'\text{.}$

##### 6

Let $G$ be an abelian group, and let $U=\{g\in G\,:\, g^{-1}=g\}.$ Prove that $U$ is a subgroup of $G\text{.}$