Articles of abstract algebra

Splicing together Short Exact Sequences

Given an exact sequence of vector spaces $$\cdots\longrightarrow V_{i-1}\longrightarrow V_{i}\longrightarrow V_{i+1}\longrightarrow\cdots$$ I want to show that it is the same as having a collection of short ones such that $$0\longrightarrow K_i \longrightarrow V_i \longrightarrow K_{i+1}\longrightarrow0$$ So to start I want to show exactness at an arbitrary $V_i$, so I space them suggestively: $$\begin{array}{c} 0&\rightarrow &K_{i-1}&\rightarrow […]

Annihilator of quotient module M/IM

Let $A$ be a commutative ring, $I$ an ideal of $A$ and $M$ an module over $A$. Is it true that $\operatorname{Ann}(M/IM) = \operatorname{Ann}(M) + I$? One inclusion is certainly true, but I don’t know about the other one. If the above statement does not hold in general, does it perhaps for finitely generated modules?

Prove $GL_2(\mathbb{Z}/2\mathbb{Z})$ is isomorphic to $S_3$

I’m asked to show that $G=GL_2(\Bbb Z/2\Bbb Z)$ is isomorphic to $S_3$. I have few ideas but I don’t manage to put them all together in order to obtain a satisying answer. I first tried using Cayley’s theorem ($G$ is isomorphic to a subgroup of $S_6$), and I also noticed that $\operatorname{Card}(G)=\operatorname{Card}(S3)=6$ & that they’re […]

Structure of $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$?

$\zeta_{15}$ is $15$th primitive $n$th root of unity. Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$ I know that if $p$ is prime then $G=Gal(\mathbb{Q}(\zeta_{p})/\mathbb{Q})=\mathbb{Z_{p}}^*$ but when $p$ is not prime, I am not show how to solve this if not prime. Does my required group have any relation to some cyclic group $\mathbb{Z_n}$, i.e. […]

Is the splitting field equal to the quotient $k/(f(x))$ for finite fields?

maybe that’s an idiot question. Given a finite field $k$ and some irreducible polynomial $f(x) \in k[x]$, then $k_f \cong k[x]/(f(x))$? I know that it’s true if $k$ is the prime field and I think that the statement is not true for general finite fields, however I could not find any counter example. Thanks in […]

Can we ascertain that there exists an epimorphism $G\rightarrow H$?

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?

Why don't we study algebraic objects with more than two operations?

Undergraduates learn about algebraic objects with one operation, namely groups, and we learn about algebraic objects with two “compatible” operations, namely rings and fields. It seems natural to then look at algebraic objects with three or more operations that are compatible, but we don’t learn about them. I asked one of my professors why this […]

One-to-one correspondence of ideals in the quotient also extends to prime ideals?

I’m beginning to learn some Grothendieck’s algebraic geometry and I have a doubt about a property of commutative algebra. For a comm. ring $A$ and an ideal $I$ of $A$, does the one-to-one correspondence between ideals of the quotient $A/I$ and ideals of $A$ containing $I$ extends to a correspondence of prime ideals ? My […]

Order of the smallest group containing all groups of order $n$ as subgroups.

Let $n\in \Bbb N$ be fixed and $m\in \Bbb N$ be the least number such that there exists a group of order $m$ in which all groups of order $n$ can be (isomorphically) embedded. Can we deduce $n!=m$?

Splitting of Automorphism Group

It is well-known that for any group $G$ there is an exact sequence $0 \rightarrow \text{Inn}(G) \rightarrow \text{Aut}(G) \rightarrow \text{Out}(G) \rightarrow 0$. Does this sequence always split, i.e. is it always true that $\text{Aut}(G)$ is a semidirect product of $\text{Inn}(G)$ and $\text{Out}(G)$? I suspect not, but have not yet come across a counterexample. Thanks in […]