ClassNumber - Maple Help

GroupTheory

 ConjugacyClass
 construct the conjugacy class of a group element
 ConjugacyClasses
 construct all the conjugacy classes of a group
 ClassNumber
 count the conjugacy classes of a group

 Calling Sequence ConjugacyClass( g, G ) ConjugacyClasses( G ) ClassNumber( G )

Parameters

 g - an element of the group G G - a group data structure or a character table

Description

 • The conjugacy class of an element $g$ of a group $G$ is the set of all conjugates ${g}^{x}={x}^{\left(-1\right)}·g·x$ for $x$ in $G$.
 • The ConjugacyClass( g, G ) command constructs the conjugacy class of an element g of a group G.
 • The group G must be an instance of a permutation group or a Cayley table group.
 • The conjugacy class of g is represented as an object cc for which the following methods are defined.

 Representative( cc ) returns the representative of the conjugacy class cc numelems( cc ) returns the number of members of the conjugacy class cc member( x, cc ) or x in cc returns true if x belongs to the conjugacy class cc Elements( cc ) returns the elements of the conjugacy class cc, as a set

 • The ConjugacyClasses( G ) command computes all the conjugacy classes of a group (or character table) G, and returns them as a set. The group G must be one for which it is possible to compute the set of all elements of G.
 • The ClassNumber( G ) command computes the number of conjugacy classes of the group (or character table) G.
 • Note that the class number of a group is a searchable property for the SearchSmallGroups command.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$

Conjugacy classes in the symmetric group are determined by the cycle type: the number of cycles of each length. So the conjugacy class in ${S}_{6}$ of permutations with one fixed point, one transposition, and one 3-cycle, contains $6\left(\begin{array}{c}6\\ 3\end{array}\right)$ elements: the support of the 3-cycle can be chosen in $\left(\begin{array}{c}6\\ 3\end{array}\right)$ ways; there are two 3-cycles given the support; and the fixed point can be chosen from the remaining three points in 3 ways. This fixes the transposition. This is verified below.

 > $\mathrm{g1}≔\mathrm{SymmetricGroup}\left(6\right)$
 ${\mathrm{g1}}{≔}{{\mathbf{S}}}_{{6}}$ (1)
 > $c≔\mathrm{ConjugacyClass}\left(\left[\left[1,2\right],\left[3,4,5\right]\right],\mathrm{g1}\right)$
 ${c}{≔}{\left(\left({1}{,}{2}\right)\left({3}{,}{4}{,}{5}\right)\right)}^{{{\mathbf{S}}}_{{6}}}$ (2)
 > $\mathrm{numelems}\left(c\right)=\mathrm{binomial}\left(6,3\right)\cdot 6$
 ${120}{=}{120}$ (3)

You can use in to iterate through the members of a conjugacy class.

 > $\mathrm{evalb}\left(\left\{\mathrm{seq}\left(x,x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}c\right)\right\}=\mathrm{Elements}\left(c\right)\right)$
 ${\mathrm{true}}$ (4)

In the following example you iterate through the members of $c$, and pair them up with their inverses if they occur. In this case, every element is conjugate with its inverse, so all elements are paired up eventually and none are left over.

 > $\mathrm{ops}≔\mathrm{Operations}\left(\mathrm{g1}\right):$
 > $\mathrm{singles}≔\varnothing :$
 > $\mathrm{pairs}≔\varnothing :$
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}c\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{inverse}≔\mathrm{ops}:-\mathrm{/}\left(x\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{if}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{inverse}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{singles}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{then}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{singles}≔\mathrm{singles}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{minus}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\mathrm{inverse}\right\};\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{pairs}≔\left\{\mathrm{op}\left(\mathrm{pairs}\right),\left\{\mathrm{inverse},x\right\}\right\}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{else}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{singles}≔\left\{x,\mathrm{op}\left(\mathrm{singles}\right)\right\}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{if}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:$
 > $\mathrm{singles}$
 ${\varnothing }$ (5)
 > $\frac{\mathrm{numelems}\left(c\right)}{\mathrm{numelems}\left(\mathrm{pairs}\right)}$
 ${2}$ (6)

Since the cycle type of a permutation on $n$ letters corresponds one-to-one with a partition $n$, the number of different conjugacy classes is equal to the partition function at $n$.

 > $\mathrm{cs1}≔\mathrm{ConjugacyClasses}\left(\mathrm{g1}\right)$
 ${\mathrm{cs1}}{≔}\left[{\left(\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}\right)\left({3}{,}{4}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}\right)\left({3}{,}{4}\right)\left({5}{,}{6}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}{,}{3}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}{,}{6}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}{,}{3}{,}{4}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}{,}{3}{,}{4}\right)\left({5}{,}{6}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}{,}{3}{,}{4}{,}{5}\right)\right)}^{{{\mathbf{S}}}_{{6}}}{,}{\left(\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}\right)\right)}^{{{\mathbf{S}}}_{{6}}}\right]$ (7)
 > $\mathrm{numelems}\left(\mathrm{cs1}\right)=\mathrm{combinat}:-\mathrm{numbpart}\left(6\right)$
 ${11}{=}{11}$ (8)

Examining the conjugacy classes of the quaternion group, given by a Cayley table.

 > $M≔\mathrm{CayleyTable}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${M}{≔}\left[\begin{array}{cccccccc}{1}& {2}& {3}& {4}& {5}& {6}& {7}& {8}\\ {2}& {3}& {4}& {1}& {7}& {5}& {8}& {6}\\ {3}& {4}& {1}& {2}& {8}& {7}& {6}& {5}\\ {4}& {1}& {2}& {3}& {6}& {8}& {5}& {7}\\ {5}& {6}& {8}& {7}& {3}& {4}& {2}& {1}\\ {6}& {8}& {7}& {5}& {2}& {3}& {1}& {4}\\ {7}& {5}& {6}& {8}& {4}& {1}& {3}& {2}\\ {8}& {7}& {5}& {6}& {1}& {2}& {4}& {3}\end{array}\right]$ (9)
 > $\mathrm{g2}≔\mathrm{CayleyTableGroup}\left(M\right)$
 ${\mathrm{g2}}{≔}{\mathrm{< a Cayley table group with 8 elements >}}$ (10)
 > $\mathrm{cs2}≔\mathrm{ConjugacyClasses}\left(\mathrm{g2}\right)$
 ${\mathrm{cs2}}{≔}\left[{{1}}^{{\mathrm{< a Cayley table group with 8 elements >}}}{,}{{2}}^{{\mathrm{< a Cayley table group with 8 elements >}}}{,}{{3}}^{{\mathrm{< a Cayley table group with 8 elements >}}}{,}{{5}}^{{\mathrm{< a Cayley table group with 8 elements >}}}{,}{{6}}^{{\mathrm{< a Cayley table group with 8 elements >}}}\right]$ (11)
 > $\mathrm{map}\left(\mathrm{print},\mathrm{map}\left(x↦'\mathrm{abs}'\left(x\right)=\mathrm{numelems}\left(x\right),\mathrm{cs2}\right)\right):$
 $\left|{{1}}^{{\mathrm{< a Cayley table group with 8 elements >}}}\right|{=}{1}$
 $\left|{{2}}^{{\mathrm{< a Cayley table group with 8 elements >}}}\right|{=}{2}$
 $\left|{{3}}^{{\mathrm{< a Cayley table group with 8 elements >}}}\right|{=}{1}$
 $\left|{{5}}^{{\mathrm{< a Cayley table group with 8 elements >}}}\right|{=}{2}$
 $\left|{{6}}^{{\mathrm{< a Cayley table group with 8 elements >}}}\right|{=}{2}$ (12)

You see that there are two conjugacy classes of size one and three of size two.

There are only two groups of order less than $512$ with class number equal to $3$.  (In fact, of any finite order.)

 > $\mathrm{cn3}≔\mathrm{SearchSmallGroups}\left('\mathrm{classnumber}'=3\right)$
 ${\mathrm{cn3}}{≔}\left[{3}{,}{1}\right]{,}\left[{6}{,}{1}\right]$ (13)

Verify the class numbers as follows:

 > $\mathrm{map}\left(\mathrm{ClassNumber}@\mathrm{SmallGroup},\left[\mathrm{cn3}\right]\right)$
 $\left[{3}{,}{3}\right]$ (14)

The class number of a direct product of groups can be computed if the class numbers of the individual factors are known.

 > $\mathrm{ClassNumber}\left(\mathrm{DirectProduct}\left(\mathrm{BabyMonster}\left(\right),\mathrm{Symm}\left(1000\right),\mathrm{ElementaryGroup}\left(19,30\right)\right)\right)$
 ${1020347182128879289726571449479665023113345049543549124072606701932879944}$ (15)

Compatibility

 • The GroupTheory[ConjugacyClass] and GroupTheory[ConjugacyClasses] commands were introduced in Maple 17.