## Every Group of Order 24 Has a Normal Subgroup of Order 4 or 8

## Problem 568

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

Add to solve laterProve that every group of order $24$ has a normal subgroup of order $4$ or $8$.

Add to solve laterLet $F$ be a finite field.

Prove that each element in the field $F$ is the sum of two squares in $F$.

Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

Add to solve later Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism.

Suppose that $f:G\to H$ is bijective.

Then there exists a map $\psi:H\to G$ such that

\[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\]
Then prove that $\psi:H \to G$ is also a group homomorphism.

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 $A=B=\Z$ be the additive group of integers.

Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$.

**(a)** Prove that $\phi$ is a group homomorphism.

**(b)** Prove that $\phi$ is injective.

**(c)** Prove that there does not exist a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi=\id_A$.

**(a)** Let $R$ be a commutative ring. If we regard $R$ as a left $R$-module, then prove that any two distinct elements of the module $R$ are linearly dependent.

**(b)** Let $f: M\to M’$ be a left $R$-module homomorphism. Let $\{x_1, \dots, x_n\}$ be a subset in $M$. Prove that if the set $\{f(x_1), \dots, f(x_n)\}$ is linearly independent, then the set $\{x_1, \dots, x_n\}$ is also linearly independent.

Read solution

Suppose that $f:R\to R’$ is a surjective ring homomorphism.

Prove that if $R$ is a Noetherian ring, then so is $R’$.

Let $f: R\to R’$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R’$.

Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.

Add to solve laterLet $f:R\to R’$ be a ring homomorphism. Let $I’$ be an ideal of $R’$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.

Add to solve later 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 $R$ be the ring of all continuous functions on the interval $[0, 2]$.

Let $I$ be the subset of $R$ defined by

\[I:=\{ f(x) \in R \mid f(1)=0\}.\]

Then prove that $I$ is an ideal of the ring $R$.

Moreover, show that $I$ is maximal and determine $R/I$.

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 $G$ 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 $R$ be a ring with unity.

Suppose that $f$ and $g$ are ring homomorphisms from $\Q$ to $R$ such that $f(n)=g(n)$ for any integer $n$.

Then prove that $f=g$.

Add to solve laterLet $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by

\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\]
where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring homomorphism, called the **augmentation map** and the kernel of $\epsilon$ is called the **augmentation ideal**.

**(a)** Prove that the augmentation ideal in the group ring $RG$ is generated by $\{g-e \mid g\in G\}$.

**(b)** Prove that if $G=\langle g\rangle$ is a finite cyclic group generated by $g$, then the augmentation ideal is generated by $g-e$.

Read solution

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 laterLet $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying

\[\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}\]
for any $g\in G$.

Let $\mu: G\times G \to G$ be a map defined by

\[\mu(g, h)=gh.\]
(That is, $\mu$ is the group operation on $G$.)

Then prove that $\phi=\mu$.

Also prove that the group $G$ is abelian.

Let $\Z$ be the ring of integers and let $R$ be a ring with unity.

Determine all the ring homomorphisms from $\Z$ to $R$.