The following table shows where isomorphism may be generally expected.
Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered.
A subgroup of a group G, which is transformed into itself by every isomorphism of G, is called a characteristic subgroup.
In the first case the isomorphism is spoken of as simple, in the second as multiple.
In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple.
isomorphism i·so·mor·phism (ī'sə-môr'fĭz'əm)
A similarity in form, as in organisms of different ancestry.
A close similarity in the crystalline structure of two or more substances of similar chemical composition.
A bijective map between two objects which preserves, in both directions, any structure under consideration. Thus a `group isomorphism' preserves group structure; an order isomorphism (between posets) preserves the order relation, and so on. Usually it is clear from context what sort of isomorphism is intended.