(Kulikov) If H and K are torsion abelian groups, prove that H ⊗Z K is a direct sum of cyclic gro

step 1 So, if x1, x2 belongs to k, then there are, there are n1, comma, n2 belonging to n, such that x

(Kulikov) If H and K are torsion abelian groups, prove that...

(Kulikov) If $H$ and $K$ are torsion abelian groups, prove that $H \otimes_{\mathrm{Z}} K$ is a direct sum of cyclic groups. Hint. Use Kulikov's Theorem: if $G$ is a $p$-primary abelian group, then there exists a pure exact sequence $0 \rightarrow B \rightarrow G \rightarrow D \rightarrow$ 0 with $B$ a direct sum of cyclic groups and $D$ divisible. Such a pure subgroup $B$ is called a basic subgroup of $G$. See Rotman, An Introduction to the Theory of Groups, p. 327 .

(Kulikov) If $H$ and $K$ are torsion abelian groups, prove that $H \otimes_{\mathrm{Z}} K$ is a direct sum of cyclic groups.
Hint. Use Kulikov's Theorem: if $G$ is a $p$-primary abelian group, then there exists a pure exact sequence $0 \rightarrow B \rightarrow G \rightarrow D \rightarrow$ 0 with $B$ a direct sum of cyclic groups and $D$ divisible. Such a pure subgroup $B$ is called a basic subgroup of $G$. See Rotman, An Introduction to the Theory of Groups, p. 327 .

Key Concepts

Pure Exact Sequence

A pure exact sequence is a short exact sequence in which the inclusion map is a pure embedding. This means that solutions of equations in the subgroup lift to solutions in the larger group, preserving a certain degree of structural compatibility. In structural decompositions of abelian groups, pure exact sequences provide a framework that ensures the decomposed components interact well with the overall group structure, such as in the formulation of basic subgroups.

Kulikov's Theorem

Kulikov's Theorem is a structural result that applies to p-primary abelian groups. It guarantees the existence of a pure exact sequence that decomposes a given group into a direct sum of a basic subgroup (which is a direct sum of cyclic groups) and a divisible part. This theorem is pivotal in identifying the cyclic components in the group and is directly used in proving that certain tensor products decompose as direct sums of cyclic groups.

Divisible Groups

Divisible groups are abelian groups in which every element is divisible by any positive integer, meaning that for every element and any integer n, there exists another element that multiplied by n gives the original element. In the context of structural theorems, divisible groups are often one of the components in a decomposition, and their properties help in concluding the overall structure of a given group.

p-primary Abelian Groups

A p-primary abelian group is a torsion abelian group in which every element has order a power of a prime number p. These groups are studied separately for each prime because their structure often exhibits regular patterns that can be exploited using specialized theorems. They play a key role in the decomposition of general torsion groups into simpler parts.

Cyclic Groups

Cyclic groups are groups that can be generated by a single element. They are the building blocks for many abelian groups, as many such groups can be decomposed into direct sums of cyclic groups. Their simple structure makes them a natural target in structural decomposition theorems.

Tensor Product of Abelian Groups

The tensor product of abelian groups is a construction that produces another abelian group from two given groups by means of a bilinear map. It is used to combine the structural properties of two groups and has important applications in homological algebra. In this context, the behavior of torsion elements in the tensor product is critical for understanding how the resulting group decomposes.

Torsion Abelian Groups

A torsion abelian group is a group in which every element has finite order. This concept is fundamental in group theory because it allows one to classify and analyze groups based on the orders of their elements. In the context of tensor products and decomposition, knowing that a group is torsion helps in applying structural theorems that simplify the analysis of its subgroups and components.

Direct Sum Decomposition

A direct sum decomposition is a way of expressing a group as a sum of subgroups such that each element of the group can be uniquely written as a combination of elements from these subgroups. This concept is crucial for understanding the internal structure of abelian groups and allows for the classification of groups by breaking them down into simpler, well-understood components like cyclic groups.

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