Articles of abstract algebra

Homomorphic image of a nilpotent group is nilpotent

Is it true that an image of a nilpotent group under a homomorphic function is nilpotent? In that case, how am I going to show this? Thanks in advance.

Prove that $\sigma G_a \sigma^{-1} = G_{\sigma(a)}$ and $\bigcap_{\sigma \in G} \sigma G_a \sigma^{-1} = 1.$

Let $G$ be a permutation group on the set $A$ (i.e.,$G \leq S_A$), let $\sigma \in G$ and let $a \in A$. Prove that $\sigma G_a \sigma^{-1} = G_{\sigma(a)}$. Deduce that if $G$ acts transitively on $A$ then $$\bigcap_{\sigma \in G} \sigma G_a \sigma^{-1} = 1.$$ Usually I know that in order to show $\sigma […]

How to show $\mathbb{Z}/\left<2,x\right>$ is isomorphic to $\mathbb{Z}_2$

I’m having quite a bit of trouble figuring out why $\mathbb{Z}[x]/\left<2,x\right>$ is isomorphic to $\mathbb{Z}_2$. So far I have figured out there is an onto map $\zeta: \mathbb{Z}\rightarrow\mathbb{Z}[x]$ given by $\zeta(n) = n$ (as a polynomial with degree 0), and that there’s another onto map $\phi: \mathbb{Z}[x]\rightarrow\mathbb{Z}[x]/\left<2,x\right>$ given by $\phi(n) = n + (x) + […]

About proving that $\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n)$

How can I prove that $$ \operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n), $$ where $\mathbb {D}_n$ is the dihedral group. Can someone help me please? Thank you.

Isomorphism of Quaternion group

Prove that $Q_{8} \cong H \rtimes G \Rightarrow H=\{e\}$ or $G=\{e\}$. My proof (is it correct?): We know that (it’s a fact from the lecture): If $M \cong H\rtimes G$, then $H \cong K \unlhd M$ and $G\cong L \leq M$. $K \cap L =\{e\}$. For finite groups: $|K||L|=|M|$. As every subgroup of $Q_{8}$ is […]

Free modules over commutative ring (possibly without unity) where free means having a LI spanning set

Let us define free module over a ring (possibly without unity) as: Def: M is said to be free module over ring R (possibly without unity) if there exist X subset of M such that X is LI and spans M. Any such X is called basis of M. Def: If M free module over […]

Confused by Example in Herstein's “Topics in Algebra”

The following comes from I.N. Herstein’s “Topics in Algebra”, just after defining subgroups. He gives the following example Let $S$ be any set and $A(S)$ be the set of one-to-one mappings of $S$ onto itself, made into a group under the composition of mappings. For any $x_0 \in S$ define $H(x_0) = \{ \phi \in […]

An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a “nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)” proving the product formula for $\phi$. How does this sequence look like? (I couldn’t figure it out myself, the term $(a,b)$ in the formula puzzles […]

$A/I$ is an injective $A/I$-module, where $A$ is PID

Suppose $A$ is PID, $I\subset A$ is a nonzero ideal, show $A/I$ is an injective $A/I$-module. I tried to prove this, but got stuck in my following argument. To show $A/I$ is injective as $A/I$-module, by the equivalence definition of injective module, we only need to show that if $M$ is an $A/I$-module such that […]

Find the Galois group

Could you help me in finding the Galois group of $\mathbb{Q}\left(\sqrt{2}, \sqrt{3}, \sqrt{(2+\sqrt{2})(3+\sqrt{3}}\right)$ over $\mathbb{Q}$? I can only say that $\mathbb{Q}(\sqrt{2}) /\mathbb{Q}$ and $\mathbb{Q}(\sqrt{3}) /\mathbb{Q}$ are two normal extentions and that $\mathbb{Q}\left(\sqrt{2}, \sqrt{3}, \sqrt{(2+\sqrt{2})(3+\sqrt{3})}\right)=\mathbb{Q}\left(\sqrt{2},\sqrt{(2+\sqrt{2})(3+\sqrt{3})}\right)$ is an eighth-degree extention over $\mathbb{Q}$. thank you! edit: Is the following a valid argument? $\mathbb{E}=\mathbb{Q}\left(\sqrt{2}, \sqrt{3} \right) \subset \mathbb{Q}\left(\sqrt{2}, \sqrt{3}, […]