equations hold.<\/p>\nStudents learning (finite) group theory often have to prove that 2 groups are isomorphic. They may construct a function from G to H, guided by their Cayley tables, then assume that the function is a homomorphism. Maybe they will check a few cases but don’t think it necessary to prove all equations hold.<\/p>\n
They are told that isomorphic groups have the same properties and, in particular, have the same number of elements of the same order. Unfortunately, they assume the converse is true which it isn’t. But the examples they see tend to confirm the converse; they don’t often see counter-examples.<\/p>\n
To make things easier let’s say two finite groups G and H are similar<\/em> if they have the same number of elements of the same order. I suspect this is entirely non-standard terminology 8-).<\/p>\n The counter-example of smallest order, 16, is where Other examples of non-isomorphic similar groups are:<\/p>\n G: \u00c2\u00a0 G: \u00c2\u00a0 Students learning (finite) group theory often have to prove that 2 groups are isomorphic. They may construct a function from G to H, guided by their Cayley tables, then assume that the function is a homomorphism. Maybe they will check a few cases but don’t think it necessary to prove all equations hold. They are […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-76","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/76","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=76"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/76\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=76"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=76"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=76"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
and 
<\/a> which are not isomorphic (G is abelian but H isn’t) but both groups have 1 element of order 1, 3 of order 2, 4 of order 4 and 8 of order 8.<\/p>\n\n
which have 
elements of order 
and 
elements of order 
<\/li>\n
. Let x be an element of order p and y, z have order q. Let 
and 
. \u00c2\u00a0G, H are the semi-direct products of Q by P with<\/li>\n
where 
\nH: \u00c2\u00a0
where 
Then G, H are non-isomorphic groups of order 
with 
elements of order 
and 
elements of order . The smallest such order is 
<\/p>\n
. Let x have order 4 and y, z order q. Let 
and . \u00c2\u00a0G, H are the semi-direct products of Q by P with<\/li>\n
\nH: \u00c2\u00a0
Then G, H are non-isomorphic groups of order 
with 
elements of order 2, 
elements of order 4 and elements of order . The smallest such order is 
.<\/ul>\n","protected":false},"excerpt":{"rendered":"