Linear algebra of finite abelian groups

If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two bases is triangular with ones on the "diagonal" (what I mean with this is explained more precisely later, in the case of abelian groups), up to possibly permuting the chosen basis of $V$. This motivates the following question, in the realm of finite abelian groups.Let $H$ be a finite abelian group with a fixed basis $h_1, \ldots, h_n$, with "basis" he...Read more

abelian groups - $p$-primary then divisible?

I asked this via MathSE, but haven't got any responces. Sorry for asking it here. Sorry.We know that in the context of abelian groups, $p$-groups are called $p$-primary groups. I have a question about $p$-primary groups as follows. Derek J.S.Robinson, noted: ...the group $\mathbb Q/\mathbb Z$ is the direct sum of its primary components, each of which is also divisible. Now the $p$-primary ...when he was giving a basic concepts and ideas of Quasicyclic Groups in chapter 4 of his book A course in the theory of groups. In another reference, An in...Read more

Irreducible characters of finite abelian groups

Let $G$ be finite abelian group and $K$ a field such that $char(K)$ does not divide the order $r$ of $G$. For each divisor $d$ of $r$ let $\omega_d$ be a primitive $d$-root of unity and $a_d:=\frac{\mid \{ a\in G \mid o(a)=d \} \mid}{dim_K(K(\omega_d))}$. By a theorem of Perlis and Walker (see e.g. Perlis/Walker) the group algebra $KG$ is isomorphic to $\bigoplus\limits_{d\mid r} K(\omega_d)^{a_d}$. Hence for each $d\mid r$ there are $a_d$ irreducible characters of dimension $dim_K(K(\omega_d))$.My question is how these irreducible characters c...Read more

lo.logic - Definable subsets of the integers as an abelian subgroup?

Consider the integers as a first-order structure in the language {0,+,-} of abelian groups. I suspect that the collection of definable subsets (without parameters) of this structure is an algebra containing nZ for all natural n (i.e. all periodic sets). This is to say, it's trivial to conclude that the collection of definable subsets contains an algebra containing nZ, however I don't know how to prove if other sets exist. Any ideas?...Read more

Minimal generation for finite abelian groups

Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,2) With orders that are powers of not necessarily distinct primes $p_1^{\alpha_1}, \ldots, p_n^{\alpha_n}$.Is it true, and how can one prove that the cardinality $c$ of any minimal generating set for $G$ satisfies $k \leq c \leq n$ (I am most concerned about the second inequality)? Here minimal means irredundant....Read more

Freeness of torsion-free abelian groups

Let $A$ be a countable torsion-free abelian group. The following conditions are well known to be equivalent:$A$ is free abelian,every finite rank pure subgroup of $A$ is free abelian.Consider the following condition:every rank one pure subgroup of $A$ is free abelian.Is this condition equivalent to the previous two? This is surely known but I was not able to (dis)prove it or find it anywhere....Read more

abelian groups - constructive Serre classes

A Serre class (of abelian groups) is a class of abelian groups closed under subgroups, quotients, and extensions. For instance, finitely generated groups and finite groups are both Serre classes.However, in constructive mathematics, these are no longer examples, at least not with the usual definition of "finite" (= in bijection with $\{0,1,\dots,n\}$ for some $n\in\mathbb{N}$). In particular, finite sets are not closed under subsets and quotients, so there is no reason that finite groups should be either.There are other weaker constructive no...Read more

Distributions over locally compact Abelian groups: when can they be Fourier transformed?

Pontryagin duality shows us every locally compact Abelian group---such as $\mathbb{R}^n$, $\mathbb{Z}$, the circle $\mathbb{R}/\mathbb{Z}$ or any finite Abelian group---has a Fourier transform. In the case of $\mathbb{R}$, it follows from distribution theory that all tempered distributions (including Dirac deltas) have Fourier transforms. I would like to know how these ideas extend to all finite Abelian groups, in particular:Question 1: how does one define distributions over locally compact Abelian groups?Question 2: what are the distributions...Read more

Showing that $\mathbb{C}^\times$ is an abelian group

QUESTIONMultiplication of complex numbers defines a binary operation on $\mathbb{C}^\times := \mathbb{C} \setminus \{0\}$. Show that $\mathbb{C}^\times$ together with this operation is an abelian group.ATTEMPTI know that for an abelian it has to show commutativity and for it to be a group there must be associativity of multiplication, but not sure where to go with it....Read more

graph isomorphism - Any two abelian group of order 8 must be isomorphic

TRUE/FALSE :Any two abelian group of order 8 must be isomorphicSOLUTION: TrueThe problem of finding all abelian groups of order 8 is impossible to solve, because there are infinitely many possibilities. But if we ask for a list of abelian groups of order 8 that comes with a guarantee that any possible abelian group of order 8 must be isomorphic to one of the groups on the list Z8, Z4 × Z2, Z2 × Z2 × Z2....Read more

About Classification of Finitely Generated Abelian Groups

I am studying Finitely Generated Abelian Groups. Now I find a material of Wolf Holzmannabelian.pdfI have a question in this material: Can I replace all notation $\oplus$ by $\times$?. More precisely, Can I replace $K\cong d_1 \mathbb{Z}\oplus \ldots \oplus d_r \mathbb{Z}$ by $K\cong d_1 \mathbb{Z}\times \ldots \times d_r \mathbb{Z}$, and $G\cong \mathbb{Z}/d_1 \mathbb{Z}\oplus \ldots \oplus \mathbb{Z}/d_r \mathbb{Z}\oplus \mathbb{Z}\oplus\ldots\oplus\mathbb{Z}$ by $G\cong \mathbb{Z}/d_1 \mathbb{Z}\times \ldots \times \mathbb{Z}/d_r \mathbb{Z}\t...Read more