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

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2
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86 views

$G$ $p$-group. If $H\triangleleft G$, then $H\cap C(G)\ne \{e\}$

I'm pretty new on this subject and I need a hint to begin to solve this question: If $G$ is a finite p-group, $H\triangleleft G$ and $H\ne \{e\}$, then $H\cap C(G)\ne \{e\}$ Thanks for any ...
3
votes
1answer
107 views

Braid Group of a Weyl Group

I am reading the paper Cherednik Algebras, Macdonald Polynomials, and Combinatorics by Mark Haiman. The definition (2.7) of the braid group $\mathcal{B}(W)$ seems to be the same as the definition of ...
1
vote
1answer
76 views

Braid invariants resource

What are some braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?
4
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2answers
119 views

Direct product of polynomial rings

Let $n = pq$, where $p$ and $q$ are distinct primes. I am trying to show that: $$\mathbb{Z}_n[X] \cong \mathbb{Z}_p[X] \times \mathbb{Z}_q[X].$$ Would it suffice to say that $\rho(np) = ...
2
votes
1answer
39 views

If $R_R$ is completely reducible then $R$ is a direct sum of finitely many fields

How can I show that if $R_R$ is completely reducible then $R$ is a direct sum of finitely many fields, when $R$ is a commutative ring? I got the reverse direction (which is true). I'm not sure in ...
8
votes
1answer
247 views

find a special element in group algebra

Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual ...
0
votes
0answers
93 views

all various cubic extensions of Q7

I need to classify all various cubic extensions of $\mathbb Q_7$? How can one do it?
2
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1answer
62 views

Submodules of an $R$-module where $R$ is the set of $n\times n$ upper triangular matrices

Let $R$ be the ring of $n\times n$ upper triangular matrices with coefficients in a field $K$. Let $V$ be the $R$-module consisting of all $1\times n$ matrices with coefficients in $K$. Define $$V_r = ...
8
votes
1answer
236 views

$S$-Units notation and Dirichlet's unit theorem

I'm having a hard time understanding some notions of a paper I'm working on. Let $L/K$ be a finite normal extension of number fields and $S$ be a set of places of $K$ prime to $p$ where $p$ denotes an ...
8
votes
2answers
250 views

Under what conditions does a ring R have the property that every zero divisor is a nilpotent element?

Under what conditions does a ring $R$ have the property that every zero divisor is a nilpotent element ? If we have a ring $R$, we know that every nilpotent element is either zero or a zero divisor. ...
4
votes
1answer
59 views

About generators of a finitely generated ideal

Let $R$ be a ring with $1$. Let $S$ be a subset of $R$, with infinitely many elements. Let $\mathfrak{i}$ be the ideal of $R$ generated by $S$. Suppose $\mathfrak{i}$ finitely generated: ...
2
votes
1answer
289 views

Homomorphism from U(30) to U(30) with a given kernel

Find a homomorphism $f$ from $U(30)$ to $U(30)$ with kernel $\{1,11\}$ and $f(7)=7$. I know $f(1)=1$ and $f(7)=7$, but not sure where to go from there.
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vote
1answer
131 views

Showing that a specific function $\mathbb Z[x]\to\mathbb Z$ is a group homomorphism.

Let $\mathbb Z[x]$ be the group of polynomials in an indeterminate $x$ with integer coefficients under addition. Prove that mapping from $\mathbb Z[x]$ into the group $\mathbb Z$ given by mapping ...
10
votes
1answer
188 views

Isomorphism between $E_8$ lattice and lattice defined by Extended Hamming Code

I have read that the following two lattices are isomorphic, and of course it seems believable, but it would be nice to have a sketch of how to construct the bijection. Let $C$ be some extended ...
3
votes
1answer
59 views

Primitve roots and congruences?

Let $p$ be an odd prime. Show that the congruence $x^4$$\equiv -1\text{ (mod }p\text{)}\ $ has a solution if and only if $p$ is of the form $8k+1$. Here is what I did Suppose that $x^4$$\equiv ...
0
votes
1answer
100 views

Let $G$ and $H$ be groups, and let $X:G\to H$ be a group homomorphism.

Let $G$ and $H$ be groups, and let $X : G\to H$ be a group homomorphism. Prove the following statements. A. If $K$ is a subgroup of $H$, then $X^{-1} (K)$ is a subgroup of $G$. B. If $K$ is normal ...
4
votes
2answers
70 views

Finite Groups: $a \in G \implies a \in H$

Let $G$ be a finite group and let $H$ be a normal subgroup. Let $a$ be an element of G and suppose that $\gcd(|a|,[G : H]) = 1$. Show that $a$ is in $H$.
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votes
2answers
564 views

Simple module is isomorphic to R/M where M is a maximal ideal

In Michael Artin's Algebra textbook page 484 Chapter 12 Exercise 1.6: A module is called simple if it is not the zero module and if it has no proper submodule. (a) Prove that any simple module is ...
-2
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1answer
97 views

Finding a primitive root modulo $13$ [duplicate]

Find a primitive root modulo each of the following integers. a) $13$ My TA said we are not going to go over this. We did not go over the topic. It seems like something good to know though. ...
1
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1answer
219 views

Finding a primitive root modulo $11^2$

Find a primitive root modulo each of the following moduli: a) $11^2$ My TA said he is not going to go over this so do not worry about it. He said you can try this if you want but he would ...
7
votes
1answer
102 views

If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$?

If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$? Could anybody clear my this doubt? Thanks for the help
2
votes
1answer
385 views

$\bigcup_{x \in G} xHx^{-1} \neq G$

I could not solve it properly: $\bigcup_{x \in G} xHx^{-1} \neq G$ if $G$ is a finite group and $H$ is a proper subgroup (of $G$). I tried to use the class equation and to create other actions to ...
4
votes
1answer
83 views

Action of $S_4$ in $S_4/S_3$

Let $G = S_4$, $H = S_3$, $X = G/H$ be the set of right cosets of $H$, $x = (14)H$ and $G $ acts on $X$ by conjugation. Compute $\mathscr{O} (x)$ and $G_x$ (the stabilizer of $x$). I've got a ...
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2answers
141 views

There is an automorphism of $\mathbb Z_6$ which is not an inner automorphism

I'm trying to show that there is an automorphism of $\mathbb Z_6$ which is not an inner automorphism. Since the generators of $\mathbb Z_6$ are 1 and 5, then we have two choices, we exclude the ...
4
votes
2answers
171 views

Proper subgroup of $\mathbb{Q}^{+}$ with finite index

Is there a non-trivial subgroup $H$ of $\mathbb{Q^{+}}$, such that $|\mathbb{Q^{+}} : H|$ is finite? Of course, $|H| = \aleph_0$, but I could not prove that such $H$ does not exist (I think it does ...
6
votes
2answers
127 views

Finding elements of order $8$ in $\mathbb{Z}_{8000000}$.

I want to find elements of order $8$ in $\mathbb{Z}_{8000000}$. I know that elements of order $n$ in a group $\mathbb{Z}_m$ is $\phi(n)$. But how can I apply this result for a group having such a ...
5
votes
3answers
158 views

Are Clifford groups very *non-commutative*?

Clifford groups seem to be very non-commutative by the relation \begin{equation} \gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}. \end{equation} But is it really so? Can we put this degree of ...
2
votes
1answer
140 views

Group of order 8 is not a simple group

Show that any Group of order 8 is not a simple group. I know that $\mathbb{Z}_8$,$\mathbb{Z}_2\times \mathbb{Z}_4$ ,$\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$,$Q_8$,$D_4$ are not simple. ...
2
votes
1answer
174 views

Categorical Confusion in Rotman's Advanced Modern Algebra Second Edition

In Rotman's Advanced Modern Algebra, exercise 6.45 (ii) in the second edition, he gives us objects $X, C_1, C_2$ and morphisms $g_1: X \rightarrow C_1$, $g_2: X \rightarrow C_2$, and asks us to prove ...
4
votes
2answers
144 views

Morphisms in a category with products

I'm having a hard time proving that $$(\psi\phi)\times(\psi\phi)=(\psi\times\psi)(\phi\times\phi),$$ where $\phi:G\to H$ and $\psi:H\to K$ in some category with products. I have seen a diagram of this ...
1
vote
2answers
64 views

Confusing with the concept of normalizer $N_G(H)$

I'm Confusing with the concept of normalizer $N_G(H)$. It's a stupid question, sorry I'm new in this subject. Following the Hungerford's concept: If $H$ acts by conjugation on the set $S$ of all ...
2
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0answers
86 views

Full Rank Matrix with a specific construction

Assume that we have a $p \times p$ matrix $Z$ over $\mathbb{F}_{2^p}$ $$Z=\begin{bmatrix} w_1 & w_1^2& w_1^4& ... & {w_1}^{2^{\frac{p}{2}-1}} & \alpha_1w_1 & ...
2
votes
1answer
87 views

how to show associativity of multiplication for not just 3 operands but for n operands

ie Id like to show a(bc)=(ab)c but for any n operands eg abcdefg=gfdcabe etc I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to ...
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votes
4answers
102 views

Allowing the zero element in a field to have an inverse

In the definition of a field one of the required properties is that every element other than zero has a multiplicative inverse. It's vague whether the zero is forced not to have an inverse or not, ...
1
vote
2answers
109 views

If $H,K \leq G$ a finite group, then $\left\lvert HK \right\rvert = \cdots$ [duplicate]

If $H,K \leq G$ a finite group, then $$\left\lvert HK \right\rvert = \frac{\left\lvert H \right\rvert \cdot \left\lvert K \right\rvert}{\left\lvert H\cap K \right\rvert}.$$ The first part of ...
1
vote
2answers
52 views

Direct Products

If $A, B, C, D$ are groups and $B\cong C \times D$. Show: $A \times B\cong A \times C \times D$ What map should I use which will help me use the given isomorphism? Do I have to define two maps? ...
4
votes
3answers
950 views

Quadratic subfield of cyclotomic field

Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
4
votes
2answers
175 views

Injective hull of $\mathbb{Z} _p$

The $p$-primary component of the group $\mathbb{Q}/\mathbb{Z}$ is denoted by $\mathbb{Z}(p^{\infty})$, where $p$ is a prime. Now I want to show that $\mathbb{Z}(p^{\infty})$ is an injective ...
1
vote
2answers
119 views

If $[G:N]$ and $|H|$ are relatively primes, then $H\lt N$.

I'm trying to show that if $N$ is a normal subgroup of $G$ and $H$ is a subgroup of $G$ such that $[G:N]$ and $|H|$ are relatively primes, then $H\lt N$. I've already found $|H|||N|$ but I couldn't go ...
4
votes
1answer
93 views

Compute the splitting field

I have to compute the splitting field of $x^6-1 \in Q[x]$. I know that $x^6-1=(x+1)(x-1)(x^2-x+1)(x^2+x+1)$ but I don't know what to do after that. Please help me.
1
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1answer
84 views

How to find the order of these groups?

I don't know why but I just cannot see how to find the orders of these groups: $YXY^{-1}=X^2$ $YXY^{-1}=X^4$ $YXY^{-1}=X^3$ With the property that $X^5 = 1$ and $Y^4 =1$ How would I go about ...
3
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0answers
81 views

Finite locally groups

Let be $V\leq \operatorname{Aut}\left( G\right),\ N\vartriangleleft G$ and $V$-invariant. Consider the semidirect product of $V$ with $G$. Let $C_{G}(V)=\{g\in G:g^{v}=g\}$ and $[G,V]=\langle ...
2
votes
2answers
137 views

Semi direct product-groups that are isomorphic

The questions asks me to find all groups up to isomorphism of the semi direct product $C_5 \rtimes C_4$ Now I've done the working out to get four groups (Note I've used $X$ as an element of $C_5$ and ...
3
votes
4answers
518 views

Prime ideal and nilpotent elements

If $\mathfrak p \subset R$ is a prime ideal, prove that for every nilpotent $r \in R$ it follows that $r \in \mathfrak p$. The only hint that my tutor gave me was to use induction. Can someone ...
0
votes
1answer
74 views

Prove that every element of $S_{n}$ can be written as a product $\sigma$ $= \tau_{1} . \tau_{2} … \tau_{r} $

Prove that every element of $S_{n}$ can be written as a product $\sigma$ $= \tau_{1} . \tau_{2} ... \tau_{r} $ of "flips"$ \\ (1,2)\ (2,3)\ ...\ (n-1,n) \ (n,1)$ of adjacent elements in the ...
6
votes
2answers
182 views

Can the associative property of a group be followed from its other properties?

Assume that we have a group together with inverse elements for all its group members, the closure property and also the identity. Can we follow from these properties that our group also has to fulfill ...
5
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0answers
89 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
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2answers
196 views

Show that r is a primitive root?

Show that if $r$ is a primitive root modulo the positive integer $m$, then $ {\overline r }$ is also a primitive root modulo m if $ {\overline r }$ is an inverse of $r$ modulo $m$. My TA did not go ...
3
votes
1answer
139 views

Galois group of irreducible quartic with real coefficients

Let $K$ be a subfield of the real numbers and $f\in K[x]$ be an irreducible quartic. If $f$ has exactly two real roots, show that the Galois group of $f$ is $S_{4}$ or $D_{4}$ (I'm using the ...
4
votes
0answers
81 views

Show that $ord(ab) | \frac{mn}{\gcd(m,n)}$ and $\frac{mn}{\gcd(m,n)^2}|ord(ab)$.

Suppose $G$ is an abelian group. Define $ord(a)=m$ and $ord(b)=n$ where $a,b \in G$. Show that $ord(ab) | \dfrac{mn}{\gcd(m,n)}$ and $\dfrac{mn}{\gcd(m,n)^2}|ord(ab)$. Since ...