## Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57

## Problem 628

Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group.

Then determine the number of elements in $G$ of order $3$.

Read solution

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Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group.

Then determine the number of elements in $G$ of order $3$.

Read solution

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

Suppose that the order $n$ of $N$ is relatively prime to the index $|G:N|=m$.

**(a)** Prove that $N=\{a\in G \mid a^n=e\}$.

**(b)** Prove that $N=\{b^m \mid b\in G\}$.

Let $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$.

Prove that the number of elements in $S$ is odd.

Add to solve later 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

\[AB=\{ab \mid a\in A, b\in B\}.\]

Let $\Q=(\Q, +)$ be the additive group of rational numbers.

**(a)** Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic.

**(b)** Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.

Let $G$ be a finite group of order $217$.

**(a)** Prove that $G$ is a cyclic group.

**(b)** Determine the number of generators of the group $G$.

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

The **product** of $H$ and $N$ is defined to be the subset

\[H\cdot N=\{hn\in G\mid h \in H, n\in N\}.\]
Prove that the product $H\cdot N$ is a subgroup of $G$.

Let $G, G’$ be groups. Let $\phi:G\to G’$ be a group homomorphism.

Then prove that for any element $g\in G$, we have

\[\phi(g^{-1})=\phi(g)^{-1}.\]

Let $G$ be a group. Suppose that we have

\[(ab)^3=a^3b^3\]
for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$.

Then prove that $G$ is an abelian group.

Add to solve later 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. 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 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 $G=\GL(n, \R)$ 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 $G$ 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 later