{"id":29,"date":"2016-09-22T20:04:57","date_gmt":"2016-09-22T20:04:57","guid":{"rendered":"http:\/\/keithmellinger.com\/magma\/?page_id=29"},"modified":"2016-11-03T19:49:14","modified_gmt":"2016-11-03T19:49:14","slug":"groups","status":"publish","type":"page","link":"https:\/\/keithmellinger.com\/magma\/groups\/","title":{"rendered":"Groups"},"content":{"rendered":"

One of the most powerful uses of Magma<\/em> is with group theory. We will only be able to touch the surface of what Magma<\/em> can do, but this should give you enough to see the power. Let\u2019s start simply. We\u2019ll define the symmetric group on 4 elements and pull out an element to examine.<\/p>\n

> S4 := SymmetricGroup(4);
\n> S4;
\nSymmetric group S4 acting on a set of cardinality 4
\nOrder = 24 = 2^3 * 3
\n> p := S4!(1,2,3);
\n> Order(p);
\n3<\/code><\/p>\n

It\u2019s easy to create subgroups generated by elements.<\/p>\n

> S := sub<S4 | p>;
\n> S;
\nPermutation group S acting on a set of cardinality 4
\nOrder = 3
\n(1, 2, 3)<\/code><\/p>\n

We can also create permutation groups by giving a list of generators. Let\u2019s look at the Klein 4 group and show that it is not cyclic.<\/p>\n

Klein4 := PermutationGroup<4 | (1,2)(3,4), (1,3)(2,4) >;
\n> #Klein4;
\n4
\n> IsAbelian(Klein4);
\ntrue
\n> IsCyclic(Klein4);
\nfalse
\n> C4 := CyclicGroup(4);
\n> Klein4,C4;
\nPermutation group Klein4 acting on a set of cardinality 4
\nOrder = 4 = 2^2
\n(1, 2)(3, 4)
\n(1, 3)(2, 4)
\nPermutation group C4 acting on a set of cardinality 4
\nOrder = 4 = 2^2
\n(1, 2, 3, 4)
\n> IsIsomorphic(Klein4,C4);
\nfalse<\/code><\/p>\n

Exercise:<\/em><\/strong> Since Magma<\/em> was written by mathematicians for mathematicians, it\u2019s often easy to guess the commands for certain things. Try to do the following without looking up the commands:<\/p>\n

    \n
  1. Use Magma<\/em> to show that the symmetric group acting on a set of cardinality 3 is the same as the dihedral group acting on a set of cardinality 3.<\/li>\n
  2. Construct a permutation group acting on 4 elements generated by the cycle elements (1,2)(3,4) and (1,2,3). Check if this group is the same as the alternating group A4<\/sub>.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    One of the most powerful uses of Magma is with group theory. We will only be able to touch the surface of what Magma can do, but this should give you enough to see the power. Let\u2019s start simply. We\u2019ll define the symmetric group on 4 elements and pull out an element to examine. > … Continue reading Groups<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-29","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/pages\/29","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/comments?post=29"}],"version-history":[{"count":5,"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/pages\/29\/revisions"}],"predecessor-version":[{"id":58,"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/pages\/29\/revisions\/58"}],"wp:attachment":[{"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/media?parent=29"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}