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GroupTheory

 Series
 description of the subgroup series data structure

Description

 • This document describes the subgroup series data structure, and the operations that can be applied to them.
 • For information about the (unrelated) power series data structure, see series.
 • A series of subgroups of a group $G$ is a nested sequence of subgroups of $G$ of either of the forms

$G={G}_{1}>{G}_{2}>\dots >{G}_{m}$

or

$1={G}_{1}<{G}_{2}<\dots <{G}_{m}$

The former is called a descending series while the latter is called an ascending series.  The subgroups

${G}_{k}$

are called terms of the series, and the integer $m$ is called its length.

 • Several commands in the GroupTheory package return series data structures representing subgroup series.  You can query the data structure for information about it by using any of the methods described below.  Particular series may support additional operations not described here.  (See the corresponding help for the particular type of series.)
 • The number of terms in a series S is returned by the numelems command with the series as argument.
 • If k is a valid index into a series, then the k-th term of the series S is gotten by the expression S[k].
 • The lowerbound( S ) command returns the least valid index into a series data structure S, and the upperbound( S ) command returns the largest valid index into S.
 • You can iterate over a series S by using either the seq command, or in a loop as shown below in the examples.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{DihedralGroup}\left(32\right)$
 ${G}{≔}{{\mathrm{D}}}_{{32}}$ (1)
 > $L≔\mathrm{LowerCentralSeries}\left(G\right)$
 ${L}{≔}{{\mathrm{D}}}_{{32}}{▹}⟨\left({1}{,}{31}{,}{29}{,}{27}{,}{25}{,}{23}{,}{21}{,}{19}{,}{17}{,}{15}{,}{13}{,}{11}{,}{9}{,}{7}{,}{5}{,}{3}\right)\left({2}{,}{32}{,}{30}{,}{28}{,}{26}{,}{24}{,}{22}{,}{20}{,}{18}{,}{16}{,}{14}{,}{12}{,}{10}{,}{8}{,}{6}{,}{4}\right)⟩{▹}{\dots }{▹}⟨\left({1}{,}{17}\right)\left({2}{,}{18}\right)\left({3}{,}{19}\right)\left({4}{,}{20}\right)\left({5}{,}{21}\right)\left({6}{,}{22}\right)\left({7}{,}{23}\right)\left({8}{,}{24}\right)\left({9}{,}{25}\right)\left({10}{,}{26}\right)\left({11}{,}{27}\right)\left({12}{,}{28}\right)\left({13}{,}{29}\right)\left({14}{,}{30}\right)\left({15}{,}{31}\right)\left({16}{,}{32}\right)⟩{▹}⟨⟩$ (2)
 > $\mathrm{numelems}\left(L\right)$
 ${6}$ (3)
 > $\mathrm{IdentifySmallGroup}\left(L\left[3\right]\right)$
 ${8}{,}{1}$ (4)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(H\right),H=L\right)$
 ${64}{,}{16}{,}{8}{,}{4}{,}{2}{,}{1}$ (5)
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}H\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}L\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{print}\left(\mathrm{IdentifySmallGroup}\left(H\right)\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:$
 ${64}{,}{52}$
 ${16}{,}{1}$
 ${8}{,}{1}$
 ${4}{,}{1}$
 ${2}{,}{1}$
 ${1}{,}{1}$ (6)

 See Also