{"id":2268,"date":"2019-05-18T15:07:22","date_gmt":"2019-05-18T22:07:22","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=2268"},"modified":"2019-05-18T15:07:22","modified_gmt":"2019-05-18T22:07:22","slug":"group-theory-for-wellness-i","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2019\/05\/group-theory-for-wellness-i\/","title":{"rendered":"Group Theory for Wellness I"},"content":{"rendered":"

(Part of Math and Wellness Month<\/a>.)<\/p>\n

Groups! A group is a set with an associative binary operation such that there exists an identity element and inverse elements! And my favorite<\/em> thing about groups is that all the time that you spend thinking about groups, is time that you're not<\/em> thinking about pain, betrayal, politics, or moral uncertainty!<\/p>\n

Groups have subgroups, which you can totally guess just from the name are subsets of the group that themselves satisfy the group axioms!<\/p>\n

The order<\/em> of a finite group is its number of elements, but this is not to be confused with the order of an element<\/em> of a group, which is the smallest integer such that the element raised to that power equals the identity! Both senses of \"order\" are indicated with vertical bars like an absolute value (|G<\/em>|, |a<\/em>|).<\/p>\n

Lagrange proved that the order of a subgroup divides the order of the group of which it is a subgroup! History remains ignorant of how often Lagrange cried.<\/p>\n

To show that a nonempty subset H<\/em> of a group is in fact a subgroup, it suffices to show that if x<\/em>, y<\/em> \u2208 H<\/em>, then xy<\/em>\u207b\u00b9 \u2208 H<\/em>.<\/p>\n

Exercise #6 in \u00a72.1 of Dummit and Foote Abstract Algebra<\/em> (3rd ed'n) asks us to prove that if G<\/em> is a commutative (\"abelian\") group, then the torsion subgroup<\/em> {g<\/em> \u2208 G<\/em> | |g| < \u221e} is in fact a subgroup. I argue as follows: we need to show that if x<\/em> and y<\/em> have finite order, then so does xy<\/em>\u207b\u00b9, that is, that (xy<\/em>\u207b\u00b9)^n<\/em> equals the identity. But (xy<\/em>\u207b\u00b9)^n<\/em> equals (xy<\/em>\u207b\u00b9)(xy<\/em>\u207b\u00b9)...(xy<\/em>\u207b\u00b9), \"n<\/em> times\"\u2014that is, pretend n<\/em> \u2265 3, and pretend that instead of \"...\" I wrote zero or more extra copies of \"(xy<\/em>\u207b\u00b9)\" so that the expression has n<\/em> factors. (I usually dislike it when authors use ellipsis notation, which feels so icky and informal compared to a nice \u03a0 or \u03a3, but let me have this one.) Because group operations are associative, we can drop the parens to get xy<\/em>\u207b\u00b9 xy<\/em>\u207b\u00b9 ... xy<\/em>\u207b\u00b9. And because we said the group was commutative, we can re\u00f6rder the factors to get xxx<\/em>...y\u207b\u00b9y\u207b\u00b9y<\/em>\u207b\u00b9, and then<\/em> we can consolidate into powers to get x<\/em>^n<\/em> y^(\u2212n<\/em>)\u2014but that's the identity if n<\/em> is the least common multiple of |x<\/em>| and |y<\/em>|, which means that xy<\/em>\u207b\u00b9 has finite order, which is what I've been trying to tell you this entire time.<\/p>\n","protected":false},"excerpt":{"rendered":"

(Part of Math and Wellness Month.) Groups! A group is a set with an associative binary operation such that there exists an identity element and inverse elements! And my favorite thing about groups is that all the time that you … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7,1],"tags":[41,77],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/2268"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=2268"}],"version-history":[{"count":1,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/2268\/revisions"}],"predecessor-version":[{"id":2269,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/2268\/revisions\/2269"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=2268"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=2268"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=2268"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}