## Prove a Group is Abelian if $(ab)^2=a^2b^2$

## Problem 401

Let $G$ be a group. Suppose that

\[(ab)^2=a^2b^2\]
for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.

Let $G$ be a group. Suppose that

\[(ab)^2=a^2b^2\]
for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.

Let $P$ be a $p$-group acting on a finite set $X$.

Let

\[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \]

The prove that

\[|X^P|\equiv |X| \pmod{p}.\]

Let $G$ be a group. Let $a$ and $b$ be elements of $G$.

If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample.

Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$.

Suppose that $G$ does not have a normal subgroup of order $3$.

Then determine all group homomorphisms from $G$ to $K$.

Let $a, b$ be relatively prime integers and let $p$ be a prime number.

Suppose that we have

\[a^{2^n}+b^{2^n}\equiv 0 \pmod{p}\]
for some positive integer $n$.

Then prove that $2^{n+1}$ divides $p-1$.

Add to solve later Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.

Let $\Aut(N)$ be the group of automorphisms of $G$.

Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.

Then prove that $N$ is contained in the center of $G$.

Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.

Prove that we have an isomorphism of groups:

\[G \cong \ker(f)\times \Z.\]

Let $H$ and $K$ be normal subgroups of a group $G$.

Suppose that $H < K$ and the quotient group $G/H$ is abelian.

Then prove that $G/K$ is also an abelian group.

Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$.

Then prove that the quotient group $G/N$ is also an abelian group.

Let $G=\GL(n, \R)$ be the **general linear group** of degree $n$, that is, the group of all $n\times n$ invertible matrices.

Consider the subset of $G$ defined by

\[\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.\]
Prove that $\SL(n, \R)$ is a subgroup of $G$. Furthermore, prove that $\SL(n,\R)$ is a normal subgroup of $G$.

The subgroup $\SL(n,\R)$ is called **special linear group**

Prove that if $G$ is a finite group of even order, then the number of elements of $G$ of order $2$ is odd.

Add to solve laterLet $G$ be a group and define a map $f:G\to G$ by $f(a)=a^2$ for each $a\in G$.

Then prove that $G$ is an abelian group if and only if the map $f$ is a group homomorphism.

Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers.

**(a)** Prove that the map $\exp:\R \to \R^{\times}$ defined by

\[\exp(x)=e^x\]
is an injective group homomorphism.

**(b)** Prove that the additive group $\R$ is isomorphic to the multiplicative group

\[\R^{+}=\{x \in \R \mid x > 0\}.\]

Let $A$ be an abelian group and let $T(A)$ denote the set of elements of $A$ that have finite order.

**(a) **Prove that $T(A)$ is a subgroup of $A$.

(The subgroup $T(A)$ is called the **torsion subgroup** of the abelian group $A$ and elements of $T(A)$ are called **torsion elements**.)

**(b)** Prove that the quotient group $G=A/T(A)$ is a **torsion-free abelian group**. That is, the only element of $G$ that has finite order is the identity element.

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

Suppose that for any non identity elements $a, b, c$ of $G$ we have

\[abc=cba. \tag{*}\]
Then prove that $G$ is an abelian group.

Let $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$.

Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.

Add to solve laterLet $G$ be a finite group. Let $a, b$ be elements of $G$.

Prove that the order of $ab$ is equal to the order of $ba$.

(Of course do not assume that $G$ is an abelian group.)

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Let $G$ be a group. (Do not assume that $G$ is a finite group.)

Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.

Let $F$ be a field and let

\[H(F)=\left\{\, \begin{bmatrix}

1 & a & b \\

0 &1 &c \\

0 & 0 & 1

\end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\]
be the **Heisenberg group** over $F$.

(The group operation of the Heisenberg group is matrix multiplication.)

Determine which matrices lie in the center of $H(F)$ and prove that the center $Z\big(H(F)\big)$ is isomorphic to the additive group $F$.

Add to solve later