A torsionless abelian group $A$ is one for which any element $a\neq 0$ can be sent to a nonzero element of $Z$ by some homomorphism $A\rightarrow Z$ (integers). Equivalently, $A$ can be embedded as a subgroup of a Cartesian power of $Z$, $Z^I$. A separable abelian group $A$ is one for which any element $a$ … Read more
Let n be a natural number. Let Un={d∈N∣d∣n and gcd(d,n/d)=1} be the set of unitary divisors, Dn be the set of divisors and Sn={d∈N∣d2∣n} be the set of square divisors of n. The set Un is a group with a⊕b:=abgcd(a,b)2. It operates on Dn via: u⊕d:=udgcd(u,d)2 The orbits of this operation “seem” to be Un⊕d=d⋅Und2 for each d∈Sn From … Read more
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 d1,d2,…,dk in such a way that di|di+1, 2) With orders that are powers of not necessarily distinct primes pα11,…,pαnn. Is it true, and how can one prove that the cardinality … Read more
Does every short exact sequence 0→Z→A→R→0 split in the category of Abelian groups? Answer The calculation of Ext1(Q,Z) can be found in this MO answer; in terms of just its isomorphism type the conclusion is that it’s an uncountable-dimensional vector space over Q, abstractly isomorphic to R. It can also be written as a quotient … Read more
Let us consider the group G:=ZaN (the product of the cyclic group with N elements with itself a times). Suppose we are given a number m that divides N. I would like to know how many elements x in G have the property that (N/m)x has order precisely m (and not any number dividing m). … Read more
For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian): finite groups ↔ finite groups discrete groups ↔ compact groups discrete torsion groups ↔ profinite groups discrete groups where each element is annhilated by some power of p ↔ pro … Read more
I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one. Are there simple formulas if one restricts to low rank for the subgroups? For example, are there formulas for enumerating cyclic subgroups, or subgroups whose minimal number … Read more
Suppose B is an abelian group such that for every integer n≥1, the n-torsion subgroup B[n] is finite. Let Btor=lim→n≥1B[n] be the torsion subgroup of B. Is it true that, necessarily, there exists an integer d≥0 such that Btor≃(Q/Z)d⊕F, for F a finite group? What if we replace Btor by B[ℓ∞]=lim→n≥1B[ℓn] for a single prime … Read more
I’ve posted this question on math.stackexchange, but haven’t gotten any responses so I’m trying here instead. Let A=Fq[T] be the ring of polynomials in one variable with coefficients in a finite field, and let r>1 be an integer. I’m currently looking for the abelianisation of the congruence subgroup Γ(N) of the special linear group SL(r,A), … Read more
Let R be an integral domain of characteristic 0 finitely generated as a ring over Z. Can the quotient group (R,+)/(Z,+) contain a divisible element? By a “divisible element” I mean an element e≠0 such that for every positive integer n there is an element f such that e=nf. As Darji points out, another way … Read more
