Normality, Quotients, and Homomorphisms

Let $H$ be a subgroup of a finite group $G$.

Let $G$ be the group of orthogonal transformations of $\mathbb {R}^{3}$ to $\mathbb {R}^{3}$ with determinant 1. Let $v \in \mathbb {R}^{3}$, $\left|v\right| = 1$, and let $H_{v} = \{ T \in G \mid Tv = v\}$.

Show that if a subgroup $H$ of a group $G$ has just one left coset different from itself then it is a normal subgroup of $G$.

Let $G$ be a group of order 120, let $H$ be a subgroup of order 24, and assume that there is at least one left coset of $H$ (other than $H$ itself) which is equal to some right coset of $H$. Prove that $H$ is a normal subgroup of $G$.

Let $G$ be a group of order 112. Prove that $G$ has a nontrivial normal subgroup.

List all groups of order 6 up to isomorphism. Find a group of order 120 that contains as a subgroup an isomorphic copy of each of them. Prove that no group of order $< 120$ has the preceding property.

For $G$ a group and $H$ a subgroup, let $C(G,H)$ denote the collection of left cosets of $H$ in $G$. Prove that if $H$ and $K$ are two subgroups of $G$ of infinite index then $G$ is not a finite union of cosets from $C(G,H) \cup C(G,K)$.

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Let $\mathbb {Z}^{2}$ be the group of lattice points in the plane (ordered pairs of integers, with coordinatewise addition as the group operation). Let $H_{1}$ be the subgroup generated by the two elements $(1,2)$ and $(4,1)$, and $H_{2}$ the subgroup generated by the two elements $(3,2)$ and $(1,3)$. Are the quotient groups $G_{1} = \mathbb {Z}^{2}/H_{1}$ and $G_{2} = \mathbb {Z}^{2}/H_{2}$ isomorphic?

Let $G$ be a group of order 10 which has a normal subgroup of order 2. Prove that $G$ is abelian.

Let $G$ be a group with three normal subgroups $N_{1}$, $N_{2}$, and $N_{3}$. Suppose $N_{i} \cap N_{j} = \{ e\}$ and $N_{i}N_{j} = G$ for all $i, j$ with $i \neq j$. Show that $G$ is abelian and $N_{i}$ is isomorphic to $N_{j}$ for all $i, j$.

Suppose $H_{i}$ is a normal subgroup of a group $G$ for $1 \leq i \leq k$, such that $H_{i} \cap H_{j} = \{ 1\}$ for $i \neq j$. Prove that $G$ contains a subgroup isomorphic to $H_{1} \times H_{2} \times \cdots \times H_{k}$ if $k = 2$, but not necessarily if $k \geq 3$.

$G$ is a group of order $n$, $H$ a proper subgroup of order $m$, and $(n/m)! < 2n$. Prove $G$ has a proper normal subgroup different from the identity.

Prove that if $G$ is a group containing no subgroup of index 2, then any subgroup of index 3 is normal.

Suppose $G$ is a nonabelian simple group, and $A$ is its automorphism group. Show that $A$ contains a normal subgroup isomorphic to $G$.

Let $G$ be a group whose order is twice an odd number. For $g$ in $G$, let $\lambda_{g}$ denote the permutation of $G$ given by $\lambda_{g}(x) = gx$ for $x \in G$.

Let $G$ be a group, $G'$ its commutator subgroup, and $N$ a normal subgroup of $G$. Suppose that $N$ is cyclic. Prove that $gn = ng$ for all $g \in G'$ and all $n \in N$.

Let $G$ be a group and $N$ be a normal subgroup of $G$ with $N \neq G$. Suppose that there does not exist a subgroup $H$ of $G$ satisfying $N \subset H \subset G$ and $N \neq H \neq G$. Prove that the index of $N$ in $G$ is finite and equal to a prime number.

Let $G$ be a group having a subgroup $A$ of finite index. Prove that there is a normal subgroup $N$ of $G$ contained in $A$ such that $N$ is of finite index in $G$.

Let $H$ be the quotient of an abelian group $G$ by a subgroup $K$. Prove or disprove each of the following statements: