Tagged Questions

4
votes
3answers
63 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
50 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 ...
7
votes
1answer
95 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
64 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
241 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
42 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
51 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 ...
7
votes
2answers
172 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
88 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
0answers
150 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. ...
17
votes
2answers
585 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 ...
4
votes
1answer
95 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
71 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
92 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
103 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 ...
0
votes
0answers
72 views

Restrictions of irreducible modules in Magma.

I am currently looking at an irreducible module M of a group G over a finite field K associated to a given representation of G. The representation maps the generators x,y of G to the matrices a,b. I ...
1
vote
2answers
75 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 ...
9
votes
2answers
618 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 ...