# Tagged Questions

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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### Whether a given algebra is the algebra of endomorphisms for a vector space.

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
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### Sylow $p$-subgroups in $S_p$ and order $p$ elements

I realise this question has been asked a few times before, but I don't understand the answers. What is the number of Sylow p subgroups in S_p? Why are the order $p$ elements in $S_p$ exactly cycles ...
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### If $K\leq H\leq G$ (not necessarily finite groups). Then prove that $[G:K]=[G:H]\cdot [H:K]$

Let $K\leq H\leq G$ (not necessarily finite groups). Why do we have $[G:K]=[G:H]\cdot [H:K]$? I can't figure out a proof in the setting of possibly infinite groups and non-normal subgroups.
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### Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
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### $G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
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### Prime ideal of $\Bbb Z[X]$ is $(0)$ or $(f)$ or $(p, f)$

Prime ideal of $\Bbb Z[X]$ is $(0)$ or $(f)$ or $(p, f)$ ($f\in\Bbb Z[X]$, $f$ is an irreducible element. And $p$ is a prime number) By the statement above, $(x^3+2,2x^2+3)$ is not a prime ideal. ...
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### What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
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### Infinitely many primes $\equiv 3 \mod 4$

Question 1 Is the following proof of the infinitude of primes $\equiv 1\mod 4$ okay? Consider a prime divisor $p\mid (n!)^2+1$. Then $(n!)^2\equiv -1 \mod p$, hence $n!$ has multiplicative order $4$ ...
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### Is $\Bbb Z[i]$ a Euclidean ring? [duplicate]

Is $\Bbb Z[i]$ a Euclidean ring? If not, what would be the simplest way of seeing that $\Bbb Z[i]$ is a PID?
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### Given a commutative ring $R$ and a monic polynomial $p(x) \in R[x]$ is $R[x]/\langle p(x) \rangle$ always a finite integral extension of $R$?

I suspect this to be true based on the fact that $p(x)$ is monic, so it should be the case that $R[x]/\langle p(x) \rangle$ is a finitely generated module over $R$, but I have no good reference for ...
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### Show that $A_5$ (alternating group of degree $n$) has $24$ elements of order $5$, $20$ elements of order $3$, and $15$ elements of order $2$.

Show that $A_5$ (alternating group of degree $n$) has $24$ elements of order $5$, $20$ elements of order $3$, and $15$ elements of order $2$. I'm a little confused on this. Wouldn't there be $5!$ ...
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### The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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### Problem with proof of $H \cap K$ is of finite index if $H,K$ are finite index subgroups

I came across a proof earlier for a solution to a Herstein Topics in Algebra question earlier that I'm not convinced with, from AOPS site. If $G$ is a group and $H,K$ are two subgroups of finite ...