# Category: Group Theory

Group Theory Problems and Solutions.

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## Problem 575

Let $G$ be a finite group of order $2n$.
Suppose that exactly a half of $G$ consists of elements of order $2$ and the rest forms a subgroup.
Namely, suppose that $G=S\sqcup H$, where $S$ is the set of all elements of order in $G$, and $H$ is a subgroup of $G$. The cardinalities of $S$ and $H$ are both $n$.

Then prove that $H$ is an abelian normal subgroup of odd order.

## Problem 568

Prove that every group of order $24$ has a normal subgroup of order $4$ or $8$.

## Problem 566

Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.

## Problem 557

Let $N$ be a normal subgroup of a group $G$.
Suppose that $G/N$ is an infinite cyclic group.

Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$.

## Problem 554

Let $x, y$ be generators of a group $G$ with relation
\begin{align*}
xy^2=y^3x,\tag{1}\\
yx^2=x^3y.\tag{2}
\end{align*}

Prove that $G$ is the trivial group.

## Problem 544

Let $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$.

Prove that the product $HK$ can never be a subgroup of the group $G$.

## Problem 523

Let $G$ be a nilpotent group and let $H$ be a proper subgroup of $G$.

Then prove that $H \subsetneq N_G(H)$, where $N_G(H)$ is the normalizer of $H$ in $G$.

## Problem 522

Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is,
$H=\{ a\in G \mid \text{the order of a is finite}\}.$

Prove that $H$ is a subgroup of $G$.

## Problem 510

Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers.

Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups.

## Problem 497

Let $G$ be an abelian group.
Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively.
Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$.

Also determine whether the statement is true if $G$ is a non-abelian group.

## Problem 495

Prove that every finite group having more than two elements has a nontrivial automorphism.

(Michigan State University, Abstract Algebra Qualifying Exam)

## Problem 493

Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying
$|A|+|B| > |G|.$ Here $|X|$ denotes the cardinality (the number of elements) of the set $X$.
Then prove that $G=AB$, where
$AB=\{ab \mid a\in A, b\in B\}.$

## Problem 490

Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$.

Define a map $\tilde{f}:H\to K$ as follows.
For each $h\in H$, there exists $g\in G$ such that $\pi(g)=h$ since $\pi:G\to H$ is surjective.
Define $\tilde{f}:H\to K$ by $\tilde{f}(h)=f(g)$.

(a) Prove that the map $\tilde{f}:H\to K$ is well-defined.

(b) Prove that $\tilde{f}:H\to K$ is a group homomorphism.

## Problem 488

Let $G$ be a finite group and let $S$ be a non-empty set.
Suppose that $G$ acts on $S$ freely and transitively.
Prove that $|G|=|S|$. That is, the number of elements in $G$ and $S$ are the same.

## Problem 474

Prove that every finite group of order $72$ is not a simple group.

## Problem 470

Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.
Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.
Then prove that $H$ is a normal subgroup of $G$.

(Michigan State University, Abstract Algebra Qualifying Exam)

## Problem 469

Let $H$ be a subgroup of a group $G$.
Suppose that for each element $x\in G$, we have $x^2\in H$.

Then prove that $H$ is a normal subgroup of $G$.

(Purdue University, Abstract Algebra Qualifying Exam)

## Problem 467

Give an example of two groups $G$ and $H$ and a subgroup $K$ of the direct product $G\times H$ such that $K$ cannot be written as $K=G_1\times H_1$, where $G_1$ and $H_1$ are subgroups of $G$ and $H$, respectively.

## Problem 465

Prove that the symmetric group $S_n$, $n\geq 3$ is a semi-direct product of the alternating group $A_n$ and the subgroup $\langle(1,2) \rangle$ generated by the element $(1,2)$.

## Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$
Let $G$ be a finite group of order $231=3\cdot 7 \cdot 11$.
Prove that every Sylow $11$-subgroup of $G$ is contained in the center $Z(G)$.