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.
Add to solve later Category: Group Theory
Add to solve later $p$-Group Acting on a Finite Set and the Number of Fixed Points
Problem 359
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
\[|X^P|\equiv |X| \pmod{p}.\]
Add to solve later Order of Product of Two Elements in a Group
Problem 354
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.
Add to solve later Group Homomorphisms From Group of Order 21 to Group of Order 49
Problem 346
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 $K$.
Add to solve later Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$.
Problem 344
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 Abelian Normal subgroup, Quotient Group, and Automorphism Group
Problem 343
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 $G$.
Add to solve later Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups
Problem 342
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:
\[G \cong \ker(f)\times \Z.\]
Add to solve later If Quotient $G/H$ is Abelian Group and $H < K \triangleleft G$, then $G/K$ is Abelian
Problem 341
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.
Add to solve later Quotient Group of Abelian Group is Abelian
Problem 340
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.
Add to solve later Special Linear Group is a Normal Subgroup of General Linear Group
Problem 332
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
Add to solve later If the Order of a Group is Even, then the Number of Elements of Order 2 is Odd
Problem 326
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 later A Group is Abelian if and only if Squaring is a Group Homomorphism
Problem 325
Let $G$ 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.
Add to solve later The Additive Group $\R$ is Isomorphic to the Multiplicative Group $\R^{+}$ by Exponent Function
Problem 322
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
\[\R^{+}=\{x \in \R \mid x > 0\}.\]
Add to solve later Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian Group
Problem 307
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.
Add to solve later If a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian Group
Problem 306
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.
Add to solve later Non-Abelian Group of Order $pq$ and its Sylow Subgroups
Problem 293
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 later The Order of $ab$ and $ba$ in a Group are the Same
Problem 291
Let $G$ 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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Add to solve later A Simple Abelian Group if and only if the Order is a Prime Number
Problem 290
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.
Add to solve later The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$
Problem 283
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 
