Irredundant Generating Sets of Finite Nilpotent Groups
نویسنده
چکیده
It is a standard fact of linear algebra that all bases of a vector space have the same cardinality, namely the dimension of the vector space over its base field. If we treat a vector space as an additive abelian group, then this is equivalent to saying that an irredundant generating set for a vector space must have cardinality equal to the dimension of the vector space. The same is not true for groups in general. For example, even a relatively simple group like Z6 can be generated by either 1 or 2 elements (〈1〉 = 〈5〉 = 〈2, 3〉 = 〈4, 3〉). This paper seeks to count the number of irredundant generating sets for direct products of elementary abelian groups, which turns out to be easily generalizable to finite nilpotent groups. We first define a function that counts partitions of a disjoint union of sets such that each block of the partition . This function allows us to break down the problem such that we need only consider the direct summands of the group. Since these turn out to be finite vector spaces, we use linear algebraic methods to study their properties. By combining formulas from each vector space with the function that counts special partitions, we are able to count irredundant generating sets of the original group.
منابع مشابه
Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup
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تاریخ انتشار 2012