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GroupTheory

 IsCPGroup
 determine whether a group is a (CP)-group
 IsCP1Group
 determine whether a group is a (CP1)-group

 Calling Sequence IsCPGroup( G ) IsCP1Group( G )

Parameters

 G - a group

Description

 • A group $G$ is a (CP)-group if each of its elements has prime-power order, where the prime may depend upon the element. A group of prime-power order is a (CP)-group, but there are (CP)-groups, such as ${\mathbf{S}}_{3}$ , whose order is divisible by more than one prime. This is equivalent to the condition that the centralizer of each non-trivial element is a $p$-group, for some prime $p$ depending upon the element. It is also equivalent for finite groups to the Gruenberg-Kegel graph of the group being totally disconnected.
 • A group $G$ is a (CP1)-group if each of its non-trivial elements has prime order where, again, the prime may depend upon the element. Equivalently, a group is a (CP1)-group if the centralizer of each non-trivial element contains only elements of order dividing $p$, for a prime $p$ depending upon the element. The symmetric group ${\mathbf{S}}_{3}$ is again an example of a (CP1)-group not of prime exponent.
 • It is a consequence of these definitions that every (CP1)-group is a (CP)-group. Both (CP)-groups and (CP1)-groups are (CN)-groups.
 • The IsCPGroup( G ) command returns true if the group G is a (CP)-group and returns false otherwise.
 • The IsCP1Group( G ) command returns true if the group G is a (CP1)-group and returns false otherwise.

Examples

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

The symmetric group ${\mathbf{S}}_{3}$ furnishes an example of a (CP)-group that is not a group of prime power order, and a (CP1)-group that is not of prime exponent.

 > $\mathrm{IsCPGroup}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (1)
 > $\mathrm{IsPGroup}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsCP1Group}\left(\mathrm{Symm}\left(3\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{Exponent}\left(\mathrm{Symm}\left(3\right)\right)$
 ${6}$ (4)

The symmetric group ${\mathbf{S}}_{4}$ is also a (CP)-group, but symmetric groups of larger degree are not. However, ${\mathbf{S}}_{4}$ is not a (CP1)-group.

 > $\mathrm{IsCPGroup}\left(\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsCP1Group}\left(\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{orseq}\left(\mathrm{IsCPGroup}\left(\mathrm{Symm}\left(n\right)\right),n=5..10\right)$
 ${\mathrm{false}}$ (7)

A cyclic group is a (CP)-group if, and only if, it has prime power order.

 > $\mathrm{seq}\left(n=\mathrm{IsCPGroup}\left(\mathrm{CyclicGroup}\left(n\right)\right),n=1..20\right)$
 ${1}{=}{\mathrm{true}}{,}{2}{=}{\mathrm{true}}{,}{3}{=}{\mathrm{true}}{,}{4}{=}{\mathrm{true}}{,}{5}{=}{\mathrm{true}}{,}{6}{=}{\mathrm{false}}{,}{7}{=}{\mathrm{true}}{,}{8}{=}{\mathrm{true}}{,}{9}{=}{\mathrm{true}}{,}{10}{=}{\mathrm{false}}{,}{11}{=}{\mathrm{true}}{,}{12}{=}{\mathrm{false}}{,}{13}{=}{\mathrm{true}}{,}{14}{=}{\mathrm{false}}{,}{15}{=}{\mathrm{false}}{,}{16}{=}{\mathrm{true}}{,}{17}{=}{\mathrm{true}}{,}{18}{=}{\mathrm{false}}{,}{19}{=}{\mathrm{true}}{,}{20}{=}{\mathrm{false}}$ (8)

And cyclic groups are (CP1)-groups precisely when the order is a prime.

 > $\mathrm{seq}\left(n=\mathrm{IsCP1Group}\left(\mathrm{CyclicGroup}\left(n\right)\right),n=1..20\right)$
 ${1}{=}{\mathrm{true}}{,}{2}{=}{\mathrm{true}}{,}{3}{=}{\mathrm{true}}{,}{4}{=}{\mathrm{false}}{,}{5}{=}{\mathrm{true}}{,}{6}{=}{\mathrm{false}}{,}{7}{=}{\mathrm{true}}{,}{8}{=}{\mathrm{false}}{,}{9}{=}{\mathrm{false}}{,}{10}{=}{\mathrm{false}}{,}{11}{=}{\mathrm{true}}{,}{12}{=}{\mathrm{false}}{,}{13}{=}{\mathrm{true}}{,}{14}{=}{\mathrm{false}}{,}{15}{=}{\mathrm{false}}{,}{16}{=}{\mathrm{false}}{,}{17}{=}{\mathrm{true}}{,}{18}{=}{\mathrm{false}}{,}{19}{=}{\mathrm{true}}{,}{20}{=}{\mathrm{false}}$ (9)

Dihedral groups are (CP)-groups just when the degree is a prime power.

 > $\mathrm{seq}\left(n=\mathrm{IsCPGroup}\left(\mathrm{DihedralGroup}\left(n\right)\right),n=1..20\right)$
 ${1}{=}{\mathrm{true}}{,}{2}{=}{\mathrm{true}}{,}{3}{=}{\mathrm{true}}{,}{4}{=}{\mathrm{true}}{,}{5}{=}{\mathrm{true}}{,}{6}{=}{\mathrm{false}}{,}{7}{=}{\mathrm{true}}{,}{8}{=}{\mathrm{true}}{,}{9}{=}{\mathrm{true}}{,}{10}{=}{\mathrm{false}}{,}{11}{=}{\mathrm{true}}{,}{12}{=}{\mathrm{false}}{,}{13}{=}{\mathrm{true}}{,}{14}{=}{\mathrm{false}}{,}{15}{=}{\mathrm{false}}{,}{16}{=}{\mathrm{true}}{,}{17}{=}{\mathrm{true}}{,}{18}{=}{\mathrm{false}}{,}{19}{=}{\mathrm{true}}{,}{20}{=}{\mathrm{false}}$ (10)

And, dihedral groups of prime degree are the only ones that are (CP1)-groups.

 > $\mathrm{seq}\left(n=\mathrm{IsCP1Group}\left(\mathrm{DihedralGroup}\left(n\right)\right),n=1..20\right)$
 ${1}{=}{\mathrm{true}}{,}{2}{=}{\mathrm{true}}{,}{3}{=}{\mathrm{true}}{,}{4}{=}{\mathrm{false}}{,}{5}{=}{\mathrm{true}}{,}{6}{=}{\mathrm{false}}{,}{7}{=}{\mathrm{true}}{,}{8}{=}{\mathrm{false}}{,}{9}{=}{\mathrm{false}}{,}{10}{=}{\mathrm{false}}{,}{11}{=}{\mathrm{true}}{,}{12}{=}{\mathrm{false}}{,}{13}{=}{\mathrm{true}}{,}{14}{=}{\mathrm{false}}{,}{15}{=}{\mathrm{false}}{,}{16}{=}{\mathrm{false}}{,}{17}{=}{\mathrm{true}}{,}{18}{=}{\mathrm{false}}{,}{19}{=}{\mathrm{true}}{,}{20}{=}{\mathrm{false}}$ (11)

Groups of prime exponent are (CP1)-groups.

 > $G≔\mathrm{SmallGroup}\left(81,12\right):$
 > $\mathrm{Exponent}\left(G\right)$
 ${3}$ (12)
 > $\mathrm{IsCP1Group}\left(G\right)$
 ${\mathrm{true}}$ (13)
 > $G≔\mathrm{SmallGroup}\left(98,4\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right)\left({3}{,}{8}\right)\left({4}{,}{7}\right)\left({5}{,}{10}\right)\left({6}{,}{9}\right)\left({11}{,}{22}\right)\left({12}{,}{24}\right)\left({13}{,}{23}\right)\left({14}{,}{19}\right)\left({15}{,}{21}\right)\left({16}{,}{20}\right)\left({17}{,}{26}\right)\left({18}{,}{25}\right)\left({27}{,}{44}\right)\left({28}{,}{46}\right)\left({29}{,}{45}\right)\left({30}{,}{48}\right)\left({31}{,}{47}\right)\left({32}{,}{39}\right)\left({33}{,}{41}\right)\left({34}{,}{40}\right)\left({35}{,}{43}\right)\left({36}{,}{42}\right)\left({37}{,}{50}\right)\left({38}{,}{49}\right)\left({51}{,}{70}\right)\left({52}{,}{69}\right)\left({53}{,}{72}\right)\left({54}{,}{71}\right)\left({55}{,}{74}\right)\left({56}{,}{73}\right)\left({57}{,}{64}\right)\left({58}{,}{63}\right)\left({59}{,}{66}\right)\left({60}{,}{65}\right)\left({61}{,}{68}\right)\left({62}{,}{67}\right)\left({75}{,}{88}\right)\left({76}{,}{87}\right)\left({77}{,}{90}\right)\left({78}{,}{89}\right)\left({79}{,}{84}\right)\left({80}{,}{83}\right)\left({81}{,}{86}\right)\left({82}{,}{85}\right)\left({91}{,}{98}\right)\left({92}{,}{97}\right)\left({93}{,}{96}\right)\left({94}{,}{95}\right){,}\left({1}{,}{3}{,}{11}{,}{27}{,}{32}{,}{14}{,}{4}\right)\left({2}{,}{7}{,}{19}{,}{39}{,}{44}{,}{22}{,}{8}\right)\left({5}{,}{12}{,}{28}{,}{51}{,}{57}{,}{33}{,}{15}\right)\left({6}{,}{13}{,}{29}{,}{52}{,}{58}{,}{34}{,}{16}\right)\left({9}{,}{20}{,}{40}{,}{63}{,}{69}{,}{45}{,}{23}\right)\left({10}{,}{21}{,}{41}{,}{64}{,}{70}{,}{46}{,}{24}\right)\left({17}{,}{30}{,}{53}{,}{75}{,}{79}{,}{59}{,}{35}\right)\left({18}{,}{31}{,}{54}{,}{76}{,}{80}{,}{60}{,}{36}\right)\left({25}{,}{42}{,}{65}{,}{83}{,}{87}{,}{71}{,}{47}\right)\left({26}{,}{43}{,}{66}{,}{84}{,}{88}{,}{72}{,}{48}\right)\left({37}{,}{55}{,}{77}{,}{91}{,}{93}{,}{81}{,}{61}\right)\left({38}{,}{56}{,}{78}{,}{92}{,}{94}{,}{82}{,}{62}\right)\left({49}{,}{67}{,}{85}{,}{95}{,}{97}{,}{89}{,}{73}\right)\left({50}{,}{68}{,}{86}{,}{96}{,}{98}{,}{90}{,}{74}\right){,}\left({1}{,}{5}{,}{17}{,}{37}{,}{38}{,}{18}{,}{6}\right)\left({2}{,}{9}{,}{25}{,}{49}{,}{50}{,}{26}{,}{10}\right)\left({3}{,}{12}{,}{30}{,}{55}{,}{56}{,}{31}{,}{13}\right)\left({4}{,}{15}{,}{35}{,}{61}{,}{62}{,}{36}{,}{16}\right)\left({7}{,}{20}{,}{42}{,}{67}{,}{68}{,}{43}{,}{21}\right)\left({8}{,}{23}{,}{47}{,}{73}{,}{74}{,}{48}{,}{24}\right)\left({11}{,}{28}{,}{53}{,}{77}{,}{78}{,}{54}{,}{29}\right)\left({14}{,}{33}{,}{59}{,}{81}{,}{82}{,}{60}{,}{34}\right)\left({19}{,}{40}{,}{65}{,}{85}{,}{86}{,}{66}{,}{41}\right)\left({22}{,}{45}{,}{71}{,}{89}{,}{90}{,}{72}{,}{46}\right)\left({27}{,}{51}{,}{75}{,}{91}{,}{92}{,}{76}{,}{52}\right)\left({32}{,}{57}{,}{79}{,}{93}{,}{94}{,}{80}{,}{58}\right)\left({39}{,}{63}{,}{83}{,}{95}{,}{96}{,}{84}{,}{64}\right)\left({44}{,}{69}{,}{87}{,}{97}{,}{98}{,}{88}{,}{70}\right)⟩$ (14)
 > $\mathrm{IsCPGroup}\left(G\right)$
 ${\mathrm{true}}$ (15)
 > $\mathrm{IsCP1Group}\left(G\right)$
 ${\mathrm{true}}$ (16)
 > $G≔\mathrm{SmallGroup}\left(72,41\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{6}{,}{3}\right)\left({4}{,}{12}{,}{5}{,}{11}\right)\left({7}{,}{33}{,}{30}{,}{34}\right)\left({8}{,}{37}{,}{29}{,}{38}\right)\left({9}{,}{41}{,}{32}{,}{42}\right)\left({10}{,}{43}{,}{31}{,}{44}\right)\left({13}{,}{53}{,}{18}{,}{39}\right)\left({14}{,}{54}{,}{17}{,}{36}\right)\left({15}{,}{55}{,}{20}{,}{35}\right)\left({16}{,}{56}{,}{19}{,}{40}\right)\left({21}{,}{57}{,}{26}{,}{58}\right)\left({22}{,}{61}{,}{25}{,}{62}\right)\left({23}{,}{65}{,}{28}{,}{66}\right)\left({24}{,}{67}{,}{27}{,}{68}\right)\left({45}{,}{60}{,}{50}{,}{71}\right)\left({46}{,}{63}{,}{49}{,}{70}\right)\left({47}{,}{64}{,}{52}{,}{69}\right)\left({48}{,}{59}{,}{51}{,}{72}\right){,}\left({1}{,}{4}{,}{6}{,}{5}\right)\left({2}{,}{11}{,}{3}{,}{12}\right)\left({7}{,}{23}{,}{30}{,}{28}\right)\left({8}{,}{24}{,}{29}{,}{27}\right)\left({9}{,}{22}{,}{32}{,}{25}\right)\left({10}{,}{21}{,}{31}{,}{26}\right)\left({13}{,}{47}{,}{18}{,}{52}\right)\left({14}{,}{48}{,}{17}{,}{51}\right)\left({15}{,}{46}{,}{20}{,}{49}\right)\left({16}{,}{45}{,}{19}{,}{50}\right)\left({33}{,}{66}{,}{34}{,}{65}\right)\left({35}{,}{63}{,}{55}{,}{70}\right)\left({36}{,}{59}{,}{54}{,}{72}\right)\left({37}{,}{68}{,}{38}{,}{67}\right)\left({39}{,}{64}{,}{53}{,}{69}\right)\left({40}{,}{60}{,}{56}{,}{71}\right)\left({41}{,}{62}{,}{42}{,}{61}\right)\left({43}{,}{58}{,}{44}{,}{57}\right){,}\left({1}{,}{6}\right)\left({2}{,}{3}\right)\left({4}{,}{5}\right)\left({7}{,}{30}\right)\left({8}{,}{29}\right)\left({9}{,}{32}\right)\left({10}{,}{31}\right)\left({11}{,}{12}\right)\left({13}{,}{18}\right)\left({14}{,}{17}\right)\left({15}{,}{20}\right)\left({16}{,}{19}\right)\left({21}{,}{26}\right)\left({22}{,}{25}\right)\left({23}{,}{28}\right)\left({24}{,}{27}\right)\left({33}{,}{34}\right)\left({35}{,}{55}\right)\left({36}{,}{54}\right)\left({37}{,}{38}\right)\left({39}{,}{53}\right)\left({40}{,}{56}\right)\left({41}{,}{42}\right)\left({43}{,}{44}\right)\left({45}{,}{50}\right)\left({46}{,}{49}\right)\left({47}{,}{52}\right)\left({48}{,}{51}\right)\left({57}{,}{58}\right)\left({59}{,}{72}\right)\left({60}{,}{71}\right)\left({61}{,}{62}\right)\left({63}{,}{70}\right)\left({64}{,}{69}\right)\left({65}{,}{66}\right)\left({67}{,}{68}\right){,}\left({1}{,}{7}{,}{8}\right)\left({2}{,}{13}{,}{14}\right)\left({3}{,}{17}{,}{18}\right)\left({4}{,}{21}{,}{22}\right)\left({5}{,}{25}{,}{26}\right)\left({6}{,}{29}{,}{30}\right)\left({9}{,}{35}{,}{39}\right)\left({10}{,}{36}{,}{40}\right)\left({11}{,}{45}{,}{46}\right)\left({12}{,}{49}{,}{50}\right)\left({15}{,}{43}{,}{37}\right)\left({16}{,}{33}{,}{41}\right)\left({19}{,}{42}{,}{34}\right)\left({20}{,}{38}{,}{44}\right)\left({23}{,}{59}{,}{63}\right)\left({24}{,}{60}{,}{64}\right)\left({27}{,}{69}{,}{71}\right)\left({28}{,}{70}{,}{72}\right)\left({31}{,}{56}{,}{54}\right)\left({32}{,}{53}{,}{55}\right)\left({47}{,}{66}{,}{58}\right)\left({48}{,}{62}{,}{68}\right)\left({51}{,}{67}{,}{61}\right)\left({52}{,}{57}{,}{65}\right){,}\left({1}{,}{9}{,}{10}\right)\left({2}{,}{15}{,}{16}\right)\left({3}{,}{19}{,}{20}\right)\left({4}{,}{23}{,}{24}\right)\left({5}{,}{27}{,}{28}\right)\left({6}{,}{31}{,}{32}\right)\left({7}{,}{35}{,}{36}\right)\left({8}{,}{39}{,}{40}\right)\left({11}{,}{47}{,}{48}\right)\left({12}{,}{51}{,}{52}\right)\left({13}{,}{43}{,}{33}\right)\left({14}{,}{37}{,}{41}\right)\left({17}{,}{42}{,}{38}\right)\left({18}{,}{34}{,}{44}\right)\left({21}{,}{59}{,}{60}\right)\left({22}{,}{63}{,}{64}\right)\left({25}{,}{69}{,}{70}\right)\left({26}{,}{71}{,}{72}\right)\left({29}{,}{56}{,}{53}\right)\left({30}{,}{54}{,}{55}\right)\left({45}{,}{66}{,}{62}\right)\left({46}{,}{58}{,}{68}\right)\left({49}{,}{67}{,}{57}\right)\left({50}{,}{61}{,}{65}\right)⟩$ (17)
 > $\mathrm{IsCPGroup}\left(G\right)$
 ${\mathrm{true}}$ (18)
 > $\mathrm{IsCP1Group}\left(G\right)$
 ${\mathrm{false}}$ (19)
 > $G≔\mathrm{SmallGroup}\left(192,182\right)$
 ${G}{≔}{\mathrm{⟨a permutation group on 192 letters with 7 generators⟩}}$ (20)
 > $\mathrm{IsCPGroup}\left(G\right)$
 ${\mathrm{false}}$ (21)
 > $\mathrm{Index}\left(\mathrm{PCore}\left(2,G\right),G\right)$
 ${6}$ (22)

The group ${}^{2}B_{2}\left(2\right)$ is the Frobenius group of order $20$ and is a (CP)-group.

 > $\mathrm{IsCPGroup}\left(\mathrm{Suzuki2B2}\left(2\right)\right)$
 ${\mathrm{true}}$ (23)

Other Frobenius groups provide important examples of (CP)-groups.

 > $\mathrm{IsCPGroup}\left(\mathrm{FrobeniusGroup}\left(72,2\right)\right)$
 ${\mathrm{true}}$ (24)

These are all the simple (CP)-groups (M. Suzuki).

 > $\mathrm{IsCPGroup}\left(\mathrm{Alt}\left(5\right)\right)$
 ${\mathrm{true}}$ (25)
 > $\mathrm{IsCPGroup}\left(\mathrm{Suzuki2B2}\left(8\right)\right)$
 ${\mathrm{true}}$ (26)
 > $\mathrm{IsCPGroup}\left(\mathrm{Suzuki2B2}\left(32\right)\right)$
 ${\mathrm{true}}$ (27)
 > $\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(2,7\right)\right)$
 ${\mathrm{true}}$ (28)
 > $\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(2,8\right)\right)$
 ${\mathrm{true}}$ (29)
 > $\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(2,9\right)\right)$
 ${\mathrm{true}}$ (30)
 > $\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(2,17\right)\right)$
 ${\mathrm{true}}$ (31)
 > $\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(3,4\right)\right)$
 ${\mathrm{true}}$ (32)
 > $\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{GruenbergKegelGraph}\left(\mathrm{PSL}\left(3,4\right)\right)\right)$

The only simple (in fact, the only insoluble) (CP1)-group is the alternating group ${\mathbf{A}}_{5}$ .

 > $\mathrm{IsCP1Group}\left(\mathrm{Alt}\left(5\right)\right)$
 ${\mathrm{true}}$ (33)

An example of an infinite (CP)-group.

 > $G≔\mathrm{QuasicyclicGroup}\left(17\right)$
 ${G}{≔}{{ℤ}}_{{{\mathrm{17}}}^{{\mathrm{\infty }}}}$ (34)
 > $\mathrm{IsCPGroup}\left(G\right)$
 ${\mathrm{true}}$ (35)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${\mathrm{\infty }}$ (36)