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Let
and be binary algebraic structures. A bijection is an isomorphism if We say
(i.e., the groups are isomorphic. - It is a bijective Group Homomorphism
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A structural property is a property that is preserved under isomorphism.
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(Fraleigh 8.15) Let
and be groups and be a one-to-one function such that . Then the image is a Subgroup of and provides an isomorphism of with . In other words -
An automorphism is an isomorphism of a group to itself (i.e.,
) - The inner automorphism of
by is defined as Performing it is called the conjugation ofby - A conjugate subgroup of
is given by
- The inner automorphism of
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As a corollary to (Fraleigh 13.18) a mapping
can be shown to be an Isomorphism by showing the following is a homomorphism is onto.