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.

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

$\mathrm{IsCPGroup}\left(\mathrm{Symm}\left(3\right)\right)$

$\mathrm{true}$

$\mathrm{IsPGroup}\left(\mathrm{Symm}\left(3\right)\right)$

$\mathrm{false}$

$\mathrm{IsCP1Group}\left(\mathrm{Symm}\left(3\right)\right)$

$\mathrm{true}$

$\mathrm{Exponent}\left(\mathrm{Symm}\left(3\right)\right)$

$6$

$\mathrm{IsCPGroup}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsCP1Group}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{orseq}\left(\mathrm{IsCPGroup}\left(\mathrm{Symm}\left(n\right)\right)\,n=5..10\right)$

$\mathrm{false}$

$\mathrm{seq}\left(n=\mathrm{IsCPGroup}\left(\mathrm{CyclicGroup}\left(n\right)\right)\,n=1..20\right)$

$1=\mathrm{true},2=\mathrm{true},3=\mathrm{true},4=\mathrm{true},5=\mathrm{true},6=\mathrm{false},7=\mathrm{true},8=\mathrm{true},9=\mathrm{true},10=\mathrm{false},11=\mathrm{true},12=\mathrm{false},13=\mathrm{true},14=\mathrm{false},15=\mathrm{false},16=\mathrm{true},17=\mathrm{true},18=\mathrm{false},19=\mathrm{true},20=\mathrm{false}$

$\mathrm{seq}\left(n=\mathrm{IsCP1Group}\left(\mathrm{CyclicGroup}\left(n\right)\right)\,n=1..20\right)$

$1=\mathrm{true},2=\mathrm{true},3=\mathrm{true},4=\mathrm{false},5=\mathrm{true},6=\mathrm{false},7=\mathrm{true},8=\mathrm{false},9=\mathrm{false},10=\mathrm{false},11=\mathrm{true},12=\mathrm{false},13=\mathrm{true},14=\mathrm{false},15=\mathrm{false},16=\mathrm{false},17=\mathrm{true},18=\mathrm{false},19=\mathrm{true},20=\mathrm{false}$

$\mathrm{seq}\left(n=\mathrm{IsCPGroup}\left(\mathrm{DihedralGroup}\left(n\right)\right)\,n=1..20\right)$

$1=\mathrm{true},2=\mathrm{true},3=\mathrm{true},4=\mathrm{true},5=\mathrm{true},6=\mathrm{false},7=\mathrm{true},8=\mathrm{true},9=\mathrm{true},10=\mathrm{false},11=\mathrm{true},12=\mathrm{false},13=\mathrm{true},14=\mathrm{false},15=\mathrm{false},16=\mathrm{true},17=\mathrm{true},18=\mathrm{false},19=\mathrm{true},20=\mathrm{false}$

$\mathrm{seq}\left(n=\mathrm{IsCP1Group}\left(\mathrm{DihedralGroup}\left(n\right)\right)\,n=1..20\right)$

$1=\mathrm{true},2=\mathrm{true},3=\mathrm{true},4=\mathrm{false},5=\mathrm{true},6=\mathrm{false},7=\mathrm{true},8=\mathrm{false},9=\mathrm{false},10=\mathrm{false},11=\mathrm{true},12=\mathrm{false},13=\mathrm{true},14=\mathrm{false},15=\mathrm{false},16=\mathrm{false},17=\mathrm{true},18=\mathrm{false},19=\mathrm{true},20=\mathrm{false}$

$G≔\mathrm{SmallGroup}\left(81\,12\right)\:$

$\mathrm{Exponent}\left(G\right)$

$3$

$\mathrm{IsCP1Group}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{SmallGroup}\left(98\,4\right)$

$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)⟩$

$\mathrm{IsCPGroup}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsCP1Group}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{SmallGroup}\left(72\,41\right)$

$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)⟩$

$\mathrm{IsCPGroup}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsCP1Group}\left(G\right)$

$\mathrm{false}$

$G≔\mathrm{SmallGroup}\left(192\,182\right)$

$G≔\mathrm{\⟨a\; permutation\; group\; on\; 192\; letters\; with\; 7\; generators\⟩}$

$\mathrm{IsCPGroup}\left(G\right)$

$\mathrm{false}$

$\mathrm{Index}\left(\mathrm{PCore}\left(2\,G\right)\,G\right)$

$6$

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

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{IsCPGroup}\left(\mathrm{Alt}\left(5\right)\right)$

$\mathrm{true}$

$\mathrm{IsCPGroup}\left(\mathrm{Suzuki2B2}\left(8\right)\right)$

$\mathrm{true}$

$\mathrm{IsCPGroup}\left(\mathrm{Suzuki2B2}\left(32\right)\right)$

$\mathrm{true}$

$\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(2\,7\right)\right)$

$\mathrm{true}$

$\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(2\,8\right)\right)$

$\mathrm{true}$

$\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(2\,9\right)\right)$

$\mathrm{true}$

$\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(2\,17\right)\right)$

$\mathrm{true}$

$\mathrm{IsCPGroup}\left(\mathrm{PSL}\left(3\,4\right)\right)$

$\mathrm{true}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{GruenbergKegelGraph}\left(\mathrm{PSL}\left(3\,4\right)\right)\right)$

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

$\mathrm{true}$

$G≔\mathrm{QuasicyclicGroup}\left(17\right)$

$G≔{\mathbb{Z}}_{{\mathrm{17}}^{\mathrm{\infty }}}$

$\mathrm{IsCPGroup}\left(G\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(G\right)$

$\mathrm{\infty }$