Subgroups

Let $G$ be a group with identity $e$.

1. Prove that the trivial subgroup, $\{e\}$, is a subgroup of $G$.

2. Prove that $G$ is a subgroup of itself.

Let $G$ be a group, and let $g\in G$. Show that $\langle g\rangle=\{g^{k}\mid k\in\mathbb{Z}\}$ is a subgroup of $G$, called the cyclic subgroup generated by $g$.

Let $G$ be a group, $H\leq G$, and $g\in G$. Define the set $gHg^{-1}=\{ghg^{-1}\mid h\in H\}$. Prove that $gHg^{-1}\leq G$ under the group operation of $G$.

Let $G$ be the group $(\mathbb{Z}_{8},\oplus_{8})$. Let $H=\{0,4\}\leq G$.

1. Explain why any left coset of $H$ in $G$ by an element $g\in G$ is the same as the right coset of $H$ in $G$ by $g$.

2. Find all distinct left cosets of $H$ in $G$.

Let $G$ be a group, $H\leq G$, and $g_{1},g_{2}\in G$. Prove that if $g_{1}H\cap g_{2}H\neq\varnothing$ then $g_{1}H=g_{2}H$.

Let $G$ be the group $(\mathbb{Z}_{99},\oplus_{99})$. It is known $H=\{0,33,66\}$ is a subgroup of $G$. What is the index of $H$ in $G$?

Prove 1.18.

Let $G$ be a cyclic group and $H\leq G$.

1. Explain why $G/H$ is a quotient group.

2. Show that $G/H$ is cyclic.

Let $G=D_{4}$, the dihedral group of order 8. By considering the subgroup axioms, determine which of the following sets, if any, are subgroups of $G$.

1. $\{e\}$

2. $\{e,r,s\}$

3. $\{r,r^{2},r^{3}\}$

4. $\{r,r^{3},r^{4},r^{6}\}$

Let $G$ be an abelian group and $H\leq G$. Define the set

Prove that

1. $K\leq G$; and

2. $H\leq K$.

Let $G$ be a group.

1. Prove that $\mathrm{Z}(G)$ is a normal subgroup of $G$.

2. Prove that $\mathrm{Z}(G)=G$ if and only if $G$ is abelian.

3. Find the center of the group $D_{4}$.(Note: you may assume $|\mathrm{Z}(G)|<\frac{1}{2}|G|$. The reason becomes apparent with 1.11’s solution.)

Let $G$ be a group, and $H,K\leq G$. Prove or disprove the following statements.

1. $H\cap K\leq G$

2. $H\cap K\leq H$

3. $H\cup K\leq G$

4. $H\cup K\leq H$

Let $G$ be a group of order 1024 and let $H$ be a proper subgroup of $G$. Determine the maximum order of $H$. Give an example of the groups $G$ and $H$ such that $H$ has this maximum order.

Let $G$ be a finite group with even order. Show that there exists an element with order 2 in $G$.

Let $G$ be a cyclic group with generator $g$. Prove that any subgroup of $G$ must also be cyclic.(Hint: Consider the solution for 1.2.)

Let $G$ be a finite group. Suppose $H$ is a subgroup of $G$ such that $[G:H]=2$. Prove that

1. $H\lhd G$;

2. $H$ contains the squares of all elements of $G$; and

3. an element $x\in G$ is also in $H$ if $x$ has odd order.

Let $G$ be a finite group with subgroups $H$ and $G$. Suppose the order of $H$ and the order of $K$ are coprime.

1. Show that the intersection of the subgroups $H$ and $K$ contains only the identity.

2. Show that, if $H$ and $K$ are normal subgroups of $G$, then for any $h\in H$ and $k\in K$ we have $hk=kh$.

Let $G$ be a finite group with order $m$.

1. State the smallest value of $m$ such that $G$ is non-abelian.

2. Prove that the value of $m$ found in (a) is the smallest value that allows $G$ to be non-abelian.

3. Hence prove that for all even integers $n\geq m$, there exists a non-abelian group of order $n$.

Let $G$ be a group and suppose $G/\mathrm{Z}(G)$ is cyclic. Prove that $G$ is abelian.