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abstract algebra

asked 2013-11-18 13:19:38 -0500

anonymous user

Anonymous

updated 2013-11-18 14:13:32 -0500

ppurka gravatar image

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.

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answered 2013-11-20 18:39:40 -0500

There 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?

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Asked: 2013-11-18 13:19:38 -0500

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Last updated: Nov 20 '13