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# Notation and Background

DOI link for Notation and Background

Notation and Background book

# Notation and Background

DOI link for Notation and Background

Notation and Background book

## ABSTRACT

The following are general facts and notations that will be used without explicit reference. By ord a we denote the order of the element a. The exponent exp G of the group G is the smallest positive integer e such that eG = 0. If nG = 0, then expG divides n. The m axim al torsion subgroup of a group G is denoted by tor G. We frequently use the Dedekind Identity or Modular Law which is true for arbitrary modules: If N < M and K is any module, then

(1.1.1)

A useful and attractive version of the Dedekind Identity is the following short exact sequence with natural maps

where M, N, K are modules and N < M. Frequently we will be considering direct decompositions Μ = K 0 L. If one

summand, say K , is kept fixed, then the possible complementary summands can be conveniently surveyed. This is the content of the following lemma which is easily verified and can be found in [Mad65, 2.6].