Editing Subgroup growth (section)

In [[mathematics]], '''subgroup growth''' is a branch of [[group theory]], dealing with quantitative questions about [[subgroup]]s of a given [[group (mathematics)|group]].<ref>{{cite book|title=Subgroup Growth|author=[[Alexander Lubotzky]], Dan Segal|year=2003|publisher=Birkhäuser|isbn=3-7643-6989-2}}</ref>

Let <math>G</math> be a [[finitely generated group]]. Then, for each integer <math>n</math> define <math>a_n(G)</math> to be the number of subgroups <math>H</math> of [[Index of a subgroup|index]] <math>n</math> in <math>G</math>. Similarly, if <math>G</math> is a [[topological group]], <math>s_n(G)</math> denotes the number of open subgroups <math>U</math> of index <math>n</math> in <math>G</math>. One similarly defines  <math>m_n(G)</math> and <math>s_n^\triangleleft(G)</math> to denote the number of [[maximal subgroup|maximal]] and [[normal subgroup]]s of index <math>n</math>, respectively.

'''Subgroup growth''' studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions.

The theory was motivated by the desire to enumerate finite groups of given order, and the analogy with [[Mikhail Gromov (mathematician)|Mikhail Gromov]]'s notion of [[Gromov's theorem on groups of polynomial growth|word growth]].