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isomorphic groups (Definition)

Two groups $ (X_1,\,*_1)$ and $ (X_2,\,*_2)$ are said to be isomorphic if there is a group isomorphism $ \psi\colon X_1\to X_2$.

Next we name a few necessary conditions for two groups $ X_1,\,X_2$ to be isomorphic (with isomorphism $ \psi$ as above).

  1. If two groups are isomorphic, then they have the same cardinality. Indeed, an isomorphism is in particular a bijection of sets.
  2. If the group $ X_1$ has an element $ g$ of order $ n$, then the group $ X_2$ must have an element of the same order. If there is an isomorphism $ \psi$ then $ \psi(g)\in X_2$ and $ (\psi(g))^n=\psi(g^n)=\psi(e_1)=e_2$ where $ e_i$ is the identity elements of $ X_i$. Moreover, if $ (\psi(g))^m=e_2$ then $ \psi(g^m)=e_2$ and by the injectivity of $ \psi$ we must have $ g^m=e_1$ so $ n$ divides $ m$. Therefore the order of $ \psi(g)$ is $ n$.
  3. If one group is cyclic, the other one must be cyclic too. Suppose $ X_1$ is cyclic generated by an element $ g$. Then it is easy to see that $ X_2$ is generated by the element $ \psi(g)$. Also if $ X_1$ is finitely generated, then $ X_2$ is finitely generated as well.
  4. If one group is abelian, the other one must be abelian as well. Indeed, suppose $ X_2$ is abelian. Then
    $\displaystyle \psi(g*_1 h)=\psi(g)*_2 \psi(h)=\psi(h)*_2 \psi(g) =\psi(h*_1 g)$
    and using the injectivity of $ \psi$ we conclude $ g*_1 h=h*_1 g$.

Note. Isomorphic groups are sometimes said to be abstractly identical, because their “abstract” structures are completely similar -- one may think that their elements are the same but have only different names.



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"isomorphic groups" is owned by alozano. [ full author list (4) | owner history (3) ]
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Also defines:  isomorphic, abstractly identical
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Cross-references: similar, abelian, finitely generated, easy to see, generated by, cyclic, divides, identity elements, order, bijection, cardinality, isomorphism, necessary, group isomorphism, groups
There are 62 references to this entry.

This is version 7 of isomorphic groups, born on 2003-10-15, modified 2006-01-11.
Object id is 5127, canonical name is IsomorphicGroups.
Accessed 7457 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
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