The RandomElement( G ) command returns a randomly selected element of the group G.

The RandomInvolution( G ) command returns a randomly selected involution (element of order $2$) of the group G. An exception is raised if G has odd order (as in that case, G contains no involutions).

For a prime number p, the RandomPElement( p, G ) command returns a random element of the group G with order equal to a power of p. An exception is raised in case p does not divide the order of G.

For a prime number p, the RandomPPrimeElement( p, G ) command returns a random element of the group G with order relatively prime to p. An exception is raised if G is a $p$-group.

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

$G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]\right)\right\}\right)$

$G≔⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

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

$\left(1\,2\right)\left(4\,5\right)$

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

$\left(1\,3\,2\right)$

$G≔\mathrm{ExceptionalGroup}\left(''G2(3)''\right)$

$G≔G_2\left(3\right)$

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

$\left(1\,184\right)\left(2\,325\right)\left(3\,144\right)\left(4\,127\right)\left(5\,207\right)\left(6\,121\right)\left(7\,193\right)\left(8\,106\right)\left(9\,238\right)\left(10\,294\right)\left(11\,13\right)\left(12\,326\right)\left(14\,197\right)\left(15\,303\right)\left(17\,267\right)\left(18\,67\right)\left(20\,107\right)\left(21\,211\right)\left(22\,219\right)\left(23\,318\right)\left(24\,161\right)\left(25\,209\right)\left(26\,126\right)\left(27\,333\right)\left(28\,256\right)\left(29\,298\right)\left(30\,44\right)\left(31\,316\right)\left(32\,194\right)\left(33\,324\right)\left(34\,250\right)\left(35\,172\right)\left(36\,78\right)\left(37\,304\right)\left(38\,321\right)\left(39\,82\right)\left(40\,117\right)\left(41\,91\right)\left(42\,276\right)\left(43\,224\right)\left(45\,47\right)\left(46\,135\right)\left(48\,280\right)\left(49\,143\right)\left(50\,344\right)\left(51\,168\right)\left(52\,96\right)\left(53\,166\right)\left(54\,165\right)\left(55\,277\right)\left(56\,230\right)\left(57\,292\right)\left(58\,338\right)\left(59\,313\right)\left(60\,334\right)\left(61\,158\right)\left(62\,237\right)\left(63\,100\right)\left(64\,89\right)\left(65\,262\right)\left(66\,195\right)\left(68\,315\right)\left(69\,71\right)\left(70\,233\right)\left(72\,263\right)\left(73\,160\right)\left(74\,131\right)\left(75\,279\right)\left(77\,154\right)\left(79\,151\right)\left(81\,124\right)\left(83\,181\right)\left(84\,268\right)\left(86\,282\right)\left(87\,141\right)\left(88\,229\right)\left(90\,191\right)\left(92\,260\right)\left(93\,108\right)\left(94\,310\right)\left(95\,152\right)\left(97\,118\right)\left(98\,252\right)\left(99\,271\right)\left(101\,192\right)\left(102\,287\right)\left(103\,203\right)\left(104\,210\right)\left(105\,272\right)\left(109\,183\right)\left(110\,347\right)\left(111\,213\right)\left(112\,142\right)\left(113\,322\right)\left(114\,115\right)\left(116\,317\right)\left(119\,284\right)\left(120\,137\right)\left(122\,288\right)\left(123\,342\right)\left(125\,218\right)\left(128\,259\right)\left(129\,157\right)\left(130\,264\right)\left(132\,349\right)\left(133\,200\right)\left(134\,198\right)\left(136\,305\right)\left(139\,330\right)\left(140\,182\right)\left(145\,270\right)\left(146\,255\right)\left(147\,240\right)\left(148\,225\right)\left(149\,351\right)\left(153\,247\right)\left(155\,169\right)\left(156\,244\right)\left(159\,306\right)\left(162\,319\right)\left(163\,320\right)\left(164\,340\right)\left(167\,314\right)\left(170\,185\right)\left(171\,220\right)\left(173\,297\right)\left(174\,269\right)\left(175\,296\right)\left(177\,302\right)\left(178\,215\right)\left(179\,285\right)\left(180\,226\right)\left(186\,335\right)\left(187\,301\right)\left(188\,234\right)\left(189\,227\right)\left(190\,293\right)\left(199\,266\right)\left(201\,241\right)\left(202\,261\right)\left(204\,254\right)\left(205\,327\right)\left(206\,251\right)\left(208\,323\right)\left(212\,236\right)\left(216\,336\right)\left(217\,308\right)\left(221\,299\right)\left(222\,295\right)\left(223\,265\right)\left(228\,311\right)\left(232\,248\right)\left(235\,291\right)\left(242\,246\right)\left(243\,286\right)\left(245\,312\right)\left(249\,253\right)\left(257\,290\right)\left(258\,307\right)\left(273\,309\right)\left(274\,283\right)\left(275\,281\right)\left(278\,300\right)\left(328\,350\right)\left(329\,346\right)\left(332\,348\right)\left(337\,341\right)\left(343\,345\right)$

$g≔\mathrm{RandomInvolution}\left(G\right)$

$g≔\left(2\,230\right)\left(3\,258\right)\left(4\,135\right)\left(5\,133\right)\left(6\,299\right)\left(7\,117\right)\left(8\,145\right)\left(9\,112\right)\left(10\,16\right)\left(11\,85\right)\left(12\,281\right)\left(13\,90\right)\left(15\,170\right)\left(18\,211\right)\left(19\,223\right)\left(20\,267\right)\left(21\,75\right)\left(22\,87\right)\left(23\,218\right)\left(24\,26\right)\left(25\,244\right)\left(27\,44\right)\left(29\,191\right)\left(30\,70\right)\left(31\,97\right)\left(32\,302\right)\left(33\,308\right)\left(34\,316\right)\left(35\,278\right)\left(36\,276\right)\left(37\,348\right)\left(38\,131\right)\left(39\,61\right)\left(40\,271\right)\left(41\,73\right)\left(42\,47\right)\left(43\,106\right)\left(45\,225\right)\left(46\,306\right)\left(48\,340\right)\left(49\,264\right)\left(50\,84\right)\left(51\,288\right)\left(52\,180\right)\left(53\,294\right)\left(54\,186\right)\left(55\,265\right)\left(56\,317\right)\left(57\,199\right)\left(58\,334\right)\left(59\,303\right)\left(60\,270\right)\left(62\,119\right)\left(63\,234\right)\left(64\,197\right)\left(65\,312\right)\left(66\,188\right)\left(67\,125\right)\left(68\,118\right)\left(69\,309\right)\left(71\,212\right)\left(72\,95\right)\left(74\,177\right)\left(76\,235\right)\left(77\,341\right)\left(78\,158\right)\left(79\,200\right)\left(80\,210\right)\left(81\,305\right)\left(82\,122\right)\left(83\,130\right)\left(86\,346\right)\left(88\,332\right)\left(89\,193\right)\left(91\,167\right)\left(92\,240\right)\left(93\,220\right)\left(94\,329\right)\left(96\,156\right)\left(98\,319\right)\left(99\,134\right)\left(100\,178\right)\left(101\,304\right)\left(102\,216\right)\left(103\,307\right)\left(104\,260\right)\left(105\,142\right)\left(107\,321\right)\left(108\,248\right)\left(109\,337\right)\left(110\,147\right)\left(111\,330\right)\left(113\,192\right)\left(114\,165\right)\left(115\,139\right)\left(116\,339\right)\left(120\,279\right)\left(121\,161\right)\left(123\,326\right)\left(124\,238\right)\left(126\,162\right)\left(127\,228\right)\left(128\,336\right)\left(129\,213\right)\left(136\,345\right)\left(138\,203\right)\left(140\,245\right)\left(141\,349\right)\left(143\,163\right)\left(144\,160\right)\left(146\,268\right)\left(148\,296\right)\left(149\,189\right)\left(150\,266\right)\left(151\,313\right)\left(152\,324\right)\left(153\,215\right)\left(154\,196\right)\left(155\,295\right)\left(157\,289\right)\left(159\,184\right)\left(164\,256\right)\left(166\,243\right)\left(168\,227\right)\left(169\,342\right)\left(173\,269\right)\left(174\,241\right)\left(175\,323\right)\left(176\,198\right)\left(179\,284\right)\left(181\,338\right)\left(182\,297\right)\left(183\,285\right)\left(185\,242\right)\left(187\,239\right)\left(190\,262\right)\left(194\,231\right)\left(195\,257\right)\left(201\,247\right)\left(202\,221\right)\left(204\,282\right)\left(205\,351\right)\left(206\,229\right)\left(207\,343\right)\left(208\,327\right)\left(214\,318\right)\left(217\,253\right)\left(219\,274\right)\left(222\,261\right)\left(224\,254\right)\left(226\,292\right)\left(232\,325\right)\left(233\,236\right)\left(246\,272\right)\left(249\,301\right)\left(250\,251\right)\left(252\,350\right)\left(259\,315\right)\left(275\,328\right)\left(277\,283\right)\left(280\,286\right)\left(287\,322\right)\left(290\,293\right)\left(291\,311\right)\left(298\,344\right)\left(300\,347\right)\left(310\,320\right)\left(331\,333\right)$

$g\mathbf{in}G$

$\mathrm{true}$

$\mathrm{PermOrder}\left(g\right)$

$2$

$G≔\mathrm{QuasicyclicGroup}\left(5\,'\mathrm{form}'=''multiplicative''\right)$

$G≔C_{5^{\mathrm{\infty }}}$

$g≔\mathrm{RandomElement}\left(G\right)$

$g≔{\ⅇ}^{\frac{2140214\mathrm{I}}{9765625}\mathrm{\pi }}$

$g\mathbf{in}G$

$\mathrm{true}$

$G≔\mathrm{DihedralGroup}\left(5\right)$

$G≔{\mathbf{D}}_5$

$g≔\mathrm{RandomPElement}\left(5\,G\right)$

$g≔\left(1\,3\,5\,2\,4\right)$

$g\mathbf{in}G$

$\mathrm{true}$

$\mathrm{PermOrder}\left(g\right)$

$5$

$g≔\mathrm{RandomPPrimeElement}\left(5\,G\right)$

$g≔\left(1\,2\right)\left(3\,5\right)$

$g\mathbf{in}G$

$\mathrm{true}$

$\mathrm{PermOrder}\left(g\right)$

$2$

$G≔\mathrm{DihedralGroup}\left(120\right)\:$

{ to 100 do PermOrder( RandomPElement( 2, G ) ) end };

$\left\{2\,4\,8\right\}$

{ to 100 do PermOrder( RandomPPrimeElement( 2, G ) ) end };

$\left\{3\,5\,15\right\}$

{ to 100 do PermOrder( RandomInvolution( G ) ) end };

$\left\{2\right\}$

The GroupTheory[RandomElement] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[RandomInvolution], GroupTheory[RandomPElement] and GroupTheory[RandomPPrimeElement] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.