ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 21 Nov 2013 01:39:40 +0100abstract algebrahttps://ask.sagemath.org/question/10743/abstract-algebra/An automorphism is an isomorphism between a group and itself. The identity
function (x -> x) is always an isomorphism, which we consider trivial. Use Sage
to construct a nontrivial automorphism of the cyclic group of order 12. Check that
the mapping is both onto and one-to-one by computing the image and kernel and
performing the proper tests on these subgroups. Now construct all of the possible
automorphisms of the cyclic group of order 12.Mon, 18 Nov 2013 20:19:38 +0100https://ask.sagemath.org/question/10743/abstract-algebra/Answer by John Palmieri for <p>An automorphism is an isomorphism between a group and itself. The identity
function (x -> x) is always an isomorphism, which we consider trivial. Use Sage
to construct a nontrivial automorphism of the cyclic group of order 12. Check that
the mapping is both onto and one-to-one by computing the image and kernel and
performing the proper tests on these subgroups. Now construct all of the possible
automorphisms of the cyclic group of order 12.</p>
https://ask.sagemath.org/question/10743/abstract-algebra/?answer=15715#post-id-15715There are (at least) three ways to define a cyclic of order 12 in Sage:
sage: C12 = groups.permutation.Cyclic(12)
sage: G12 = AbelianGroup([12])
sage: H12 = AdditiveAbelianGroup([12])
Now evaluate `C12.gens()` (for example): how many generators does this group have? Any homomorphism is determined by where the generators go. So what are the possible endomorphisms (= homomorphisms from the group to itself) of this group? Can you figure out which of them are actually automorphisms?
Thu, 21 Nov 2013 01:39:40 +0100https://ask.sagemath.org/question/10743/abstract-algebra/?answer=15715#post-id-15715