isomorphic groups
Two groups and are said to be isomorphic if there is a group isomorphism .
Next we name a few necessary conditions for two groups to be isomorphic (with isomorphism as above).
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1.
If two groups are isomorphic, then they have the same cardinality. Indeed, an isomorphism is in particular a bijection of sets.
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2.
If the group has an element of order , then the group must have an element of the same order. If there is an isomorphism then and where is the identity elements of . Moreover, if then and by the injectivity of we must have so divides . Therefore the order of is .
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3.
If one group is cyclic, the other one must be cyclic too. Suppose is cyclic generated by an element . Then it is easy to see that is generated by the element . Also if is finitely generated, then is finitely generated as well.
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4.
If one group is abelian, the other one must be abelian as well. Indeed, suppose is abelian. Then
and using the injectivity of we conclude .
Note. Isomorphic groups are sometimes said to be abstractly identical, because their “abstract” are completely similar — one may think that their elements are the same but have only different names.
Title | isomorphic groups |
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Canonical name | IsomorphicGroups |
Date of creation | 2013-03-22 14:01:58 |
Last modified on | 2013-03-22 14:01:58 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 10 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 20A05 |
Defines | isomorphic |
Defines | abstractly identical |