{"id":41,"date":"2016-09-22T20:40:35","date_gmt":"2016-09-22T20:40:35","guid":{"rendered":"http:\/\/keithmellinger.com\/magma\/?page_id=41"},"modified":"2016-11-03T19:51:24","modified_gmt":"2016-11-03T19:51:24","slug":"subgroups","status":"publish","type":"page","link":"https:\/\/keithmellinger.com\/magma\/subgroups\/","title":{"rendered":"Subgroups"},"content":{"rendered":"

Subgroups are studied in most first courses in abstract algebra. Computing subgroups of a cyclic group is pretty straight forward, but it\u2019s harder to get students to understand subgroups of a non-cyclic or, even better, a non-abelian group. With Magma<\/em>, it\u2019s easy to compute examples. Let\u2019s start with something simple.<\/p>\n

> A5 := AlternatingGroup(5); <\/code>
\n > A5; <\/code>
\n Permutation group A5 acting on a set of cardinality 5 <\/code>
\n Order = 60 = 2^2 * 3 * 5 <\/code>
\n (3, 4, 5) <\/code>
\n (1, 2, 3) <\/code><\/p>\n

Computing the subgroups of A5 is easy in Magma<\/em>. Trying executing the following command and see if you can understand what it is giving you:<\/p>\n

> Subgroups(A5); <\/code><\/p>\n

We can do the same for cyclic subgroups:<\/p>\n

> CSubs := CyclicSubgroups(A5); <\/code>
\n > CSubs; <\/code><\/p>\n

Or abelian subgroups:<\/p>\n

> ASubs := AbelianSubgroups(A5); <\/code>
\n > ASubs; <\/code><\/p>\n

Magma<\/em> will allow you to access (abstractly) one of the subgroups by choosing its location in the list followed by \u201c`subgroup\u201d:<\/p>\n

> S := ASubs[5]`subgroup; <\/code>
\n > S; <\/code>
\n Permutation group S acting on a set of cardinality 5 <\/code>
\n Order = 4 = 2^2 <\/code>
\n (2, 5)(3, 4) <\/code>
\n (2, 4)(3, 5) <\/code><\/p>\n

Exercise:<\/em><\/strong> Create the dihedral group acting on an octagon. Find all the cyclic subgroups, and give a geometric description of those with order 2. Start by brainstorming for the answer and then use Magma<\/em> to confirm your beliefs. You can guess the command for generating the Dihedral Group acting on a 8-sided figure, can’t you?<\/p>\n","protected":false},"excerpt":{"rendered":"

Subgroups are studied in most first courses in abstract algebra. Computing subgroups of a cyclic group is pretty straight forward, but it\u2019s harder to get students to understand subgroups of a non-cyclic or, even better, a non-abelian group. With Magma, it\u2019s easy to compute examples. Let\u2019s start with something simple. > A5 := AlternatingGroup(5); > … Continue reading Subgroups<\/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-41","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/pages\/41","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=41"}],"version-history":[{"count":4,"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/pages\/41\/revisions"}],"predecessor-version":[{"id":59,"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/pages\/41\/revisions\/59"}],"wp:attachment":[{"href":"https:\/\/keithmellinger.com\/magma\/wp-json\/wp\/v2\/media?parent=41"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}