1
vote
0answers
20 views

Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. Is ...
2
votes
2answers
123 views

Applications of groups, rings and modules in life [closed]

I have read calculus. Now I'm reading algebra. What is the application of groups, rings and modules in life? When we study derivations we know several applications. What about groups, rings and ...
0
votes
1answer
19 views

Minimal Frattini Extensions

From "Construction of Finite Groups" article : "To construct minimal Frattini extensions of $H$ we have to consider all suitable irreducible $H$-modules $M$. Clearly, $M$ can be viewed as an ...
1
vote
1answer
38 views

Extension of a group by a module

What is the definition of an extension of a group by a module? and what is the difference between extensions by groups and modules?
2
votes
1answer
42 views

What does this notation mean, which looks like direct sum?

I ran across this notation in a paper. What does $ Z^{\bigoplus n(n-1)}$ mean? I knew that $ Z^n$ or $ Z^{(n)} $ means direct sum, but never met this one.
1
vote
1answer
37 views

What does this notation mean? Perhaps related to abelian group and module.

I found this in a paper, but I don't it's meaning. Where H is an abelian group, what does $ \bigwedge ^{2} H $ mean? If H is a dual vector space, perhaps it is wedge product. But I don't know when it ...
3
votes
1answer
53 views

Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...
0
votes
1answer
26 views

How to show a Hom group is isomorphic to some modulo group?

Let $\mathbb{Z}_{n}[m]=\{a\in{\mathbb{Z}_{n}|ma=0\}}$, where $m$ is a positive integer. Then how do we establish the following? $$ \mathbb{Z}_{n}[m]\cong{\mathbb{Z}_{\gcd(m,n)}} $$
1
vote
2answers
61 views

Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
0
votes
2answers
108 views

Possible difference between $\mathbb{Z}$-modules and vector spaces

Suppose $G$ is a free abelian groups, i.e. a free $\mathbb{Z}$-module; we have a set $S \subset G $ such that $S$ spans $G$. Can we conclude that the rank of $G$ as a $\mathbb{Z}$-module is $ \leq ...
2
votes
1answer
84 views

Understanding the Bockstein homomorphism in group cohomology

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism ...
3
votes
1answer
72 views

Maximal $\mathbb{Z}$-submodules in $\mathbb{Q}$

Is it true that $\mathbb{Q}$ viewed as $\mathbb{Z}$-module ( i.e. abelian group ) has not maximal $\mathbb{Z}$-submodules ? Why ?
2
votes
2answers
81 views

Characters of a faithful irreducible Module for an element in the centre

Basically here is my questions. We have a character $\chi$ which is faithful and irreducible of a group $G$. we have an element $g$ which i needs to show belongs to the centre $Z(G)$, i.e. $gh=hg$ for ...
1
vote
1answer
37 views

Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
0
votes
1answer
29 views

modules finite congenerated are closed under extensions

I have to prove some properties about modules finite cogenerated, I´ve already prove that mmodules finite cogenerated are closed under submodules, finite direct sums, but I can´t see how to prove that ...
2
votes
1answer
145 views

Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
6
votes
1answer
75 views

The cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$

I want to calculate the cardinality of the group $\mathrm{SL}(k,\mathbb{Z}/n\mathbb{Z})$, $k,n \in \mathbb{N}$. When $n$ is prime, it is easy to calculate, so I want to know the general case.
1
vote
1answer
37 views

A Problem about Spliting of Exact Sequence

Module Case Let $$0\rightarrow N \stackrel f \rightarrow G\stackrel g \rightarrow Q\rightarrow 0$$ be the exact $R$-module sequence iff (1) there exists $u:G\rightarrow N$ which satisfies $u\circ ...
1
vote
0answers
122 views

Isomorphism theorem for Abelian groups, related to Hatcher exercise 2.1.14

I am trying to understand which Abelian groups can fit the short exact sequence \begin{equation} 0 \rightarrow \mathbb{Z}_{p ^m}\rightarrow A \rightarrow \mathbb{Z}_{p^n}\rightarrow 0. \end{equation} ...
2
votes
1answer
44 views

How do we compute $\mathbb{Z}^2/(n, m)$?

I have been trying to compute quotients of this form (as modules). Does anyone have a quick method of doing this? I'm sure I'm missing something obvious. Any help is appreciated!
2
votes
1answer
107 views

Why are projective modules cohomologically trivial?

Let $G$ be a finite group, $H\subset G$ a subgroup, $k$ a commutative ring, $M$ a $kG$-module, $n\in\mathbb{Z}$, and $\hat{H}\,^n(H,M)$ the $n$th Tate cohomology group as defined in this question, ...
0
votes
1answer
68 views

Question about the order of an element at $\mathbb{Z}_{n}^{*}$

Assume that $n,q>1$ and $n=\frac{q^r-1}{q-1}$. I know that $q\in\mathbb{Z}_{n}^{*}$. How do I prove that $ord(q)=r$? This is the meaning of $\mathbb{Z}_{n}^{*}$: $$\mathbb{Z}_{n}^{*}= ...
1
vote
1answer
34 views

Question about an element at $\mathbb{Z}_{n}^{*}$

Assume that $n,q>1$ and $n=\frac{q^r-1}{q-1}$. How do I prove that $q\in \mathbb{Z}_{n}^{*}$? $$\mathbb{Z}_{n}^{*}= \left\{a\mid1\le a<n;\ \gcd(a,n)=1 \right\}$$ Thank you!
-1
votes
1answer
91 views

Module, vector space and abelian group

please can someone help me to prove this two properties $\bullet$ If K is a field, then the concept of K-vector space (a vector space over K) and K-module are identical. $\bullet$ The concept of a ...
0
votes
1answer
24 views

Modules over a group presented via a free group.

Say $G$ is presented via a free group $F$ freely generated by $S=\{s_i, 1=1,2,\dots\}$. Then $\pi:F \rightarrow G$ the canonical projection. Let $R$ be any commutative ring. Can we follow that any ...
1
vote
0answers
63 views

Finite $\Delta$-module of $p$-power order

I have a question concerning lemma, that I want to prove: Let $p$ be a prime and $\Delta$ be a finite group of order prime to $p$. Let $M$ be a finite $\Delta$-module of order a power of $p$. Then ...
1
vote
0answers
74 views

Extending isomorphisms in the semi-simple case.

Is there some proposition saying how to extend an isomorphism of $k$-vector spaces where $k$ is a field of characteristic $p$ to an isomorphismus of $k[H]$-modules where $H$ is a group of order prime ...
0
votes
1answer
34 views

Irreducibility of a module of a cyclic group

Let $G = \langle x \rangle$ be a cyclic group of order $p$ ($p$ is a prime). Let $M$ be a vector space over $\mathbb Q$ with basis $\{m_0, m_1, \cdots, m_{p-1}\}$. Define $\rho(x)$ to be $$\rho(x)m_i ...
1
vote
1answer
70 views

Eigenspace decomposition for semisimple module

Let's start with a prime $p$ and a group $\Delta$ of order prime to $p$. Let $M$ be a finite $\mathbb{F}_p[\Delta]$-module of order a power of $p$. I want to find a decomposition into eigenspaces ...
3
votes
3answers
137 views

Homomorphisms between $ \mathbb{Z} $ modules.

Calculate $\newcommand\Hom{\operatorname{Hom}}\Hom(\mathbb Z \oplus \mathbb Z_{p^\infty},\mathbb Z \oplus \mathbb Z_{p^\infty})$. Where $ \mathbb Z_{p^\infty}= ...
0
votes
1answer
65 views

$M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)

I was rather surprised by the fact that two modules are isomorphic if and only if their abelian group structures are isomorphic. I might just sketch the proof here. Given a ring homomorhpism ...
8
votes
1answer
454 views

Countable infinite direct product of $\mathbb{Z}$ modulo countable direct sum

Let $M=\mathbb Z^{\mathbb N}$ be the product of a countable number of copies of the group $\mathbb{Z}$ and let $N=\mathbb Z^{(\mathbb N)}$ be the direct sum of a countable number of copies of ...
2
votes
2answers
75 views

Is this function from $M_g \to \mathbb{Z} \otimes M$ well defined?

Given a $G$-module $M$ (a $\mathbb{Z}G$-module), let $M_g$ denote the group of coinvariants, that is $M/\langle m -gm \rangle$. If we regard $\mathbb{Z}$ as a right $\mathbb{Z}G$-module with trivial ...
10
votes
7answers
1k views

Applications of the Isomorphism theorems

In my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but I ...
1
vote
0answers
50 views

Is there an advantage in thinking about modules in a specific way?

In Isaacs' Algebra, before even modules are introduced, the author introduces operator groups ( http://en.wikipedia.org/wiki/Group_with_operators ) and gets many results, which can be applied to ...
1
vote
2answers
68 views

Show by example that this need not be true if we do not assume that the groups are finitely generated

Let $G, H,$ and $K$ be finitely generated abelian groups. If $G \times K \cong H \times K$, show that $G \cong H$. Show by example that this need not be true if we do not assume that the groups ...
6
votes
2answers
182 views

Why is this statement about generators of groups true?

Let $G$ be free abelian of rank $n$ and $H \subseteq G$ a subgroup also of rank $n$. It is known that $G/H$ is finite, in fact a direct sum of at most $n$ cyclic groups. Thus we can write $$G/H = ...
1
vote
1answer
142 views

Every subgroup of finite rank of $\mathbb Z^\mathbb N$ is free abelian

I want to prove that every subgroup of finite rank of $\mathbb Z^\mathbb N$ is free abelian. A group $G$ has finite rank $N$ if there exists a subset $ \{e_1,..,e_N\}$ of $G$ such that: $(1)$ If $a_i ...
1
vote
1answer
285 views

Hungerford Algebra - Chapter IV - proof of the splitting lemma

Im working with the splitting lemma right now and I don't understand it too much. If someone has that book and could help me complete the sketch in page 177, proof of 1.18) I would be very grateful. ...
19
votes
2answers
1k views

Applications of the Jordan-Hölder Theorem.

All books about group theory bring the Jordan-Hölder theorem, but does not give applications. I only saw in Rotman's book the Fundamental Theorem of Arithmetic being proved as a consequence of this ...
5
votes
1answer
162 views

Idempotent and Hermitian vectors in Group Algebra

Let $C$ be the field of complex number and $G$ a finite group, then define $C[G]$ be a vector space over $C$, with elemnts of $G$ as the basis. Then any element in $C[G]$ can be written as $\sum_{g ...
3
votes
1answer
77 views

Given a group $G$ and $H\leq G$ is there a homomorphism $f: G \rightarrow G$ s.t $f(H) = id$ and $f(x) \neq id$ if $x\notin H$?

The motivation for this question is that I am trying to Read Basic Algebra 2 (despite a somewhat weak background for the book) and if this is true then it is fairly easy to show that monics in Grp are ...
3
votes
1answer
119 views

The Frobenius-Nakayama Formula

I am currently reading a paper where it refers to the usual Frobenius-Nakayama formula describing quotients of an induced module. It is refering to the following result: If $k$ is a field, $P$ is ...
0
votes
1answer
108 views

Finding subgroups of index 2 of $G = \prod\limits_{i=1}^\infty \mathbb{Z}_n$

I looked at this question and its answer. The answer uses the fact that every vector space has a basis, so there are uncountable subgroups of index 2 if $n=p$ where $p$ is prime. Are there ...
1
vote
2answers
114 views

Prove that the group ring $\mathbb{Z}_2[G]$ is not semisimple where $G$ is the cyclic group of order 2

I want to prove that the group ring $\mathbb{Z}_2[G]$ is not semisimple when $G$ is the cyclic group of order two. I guess the simplest way would be to show that there exists a submodule that is not a ...
13
votes
2answers
1k views

Short Exact Sequences & Rank Nullity

This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise (see for example pp. 146 of Hatcher) If $0 \stackrel{id}{\to} A ...