Group homomorphism

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that h ( u ∗ v ) = h ( u ) ⋅ h ( v ) {\displaystyle h(u*v)=h(u)\cdot h(v)} where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, h ( e G ) = e H {\displaystyle h(e_{G})=e_{H}} and it also maps inverses to inverses in the sense that h ( u − 1 ) = h ( u ) − 1 . {\displaystyle h\left(u^{-1}\right)=h(u)^{-1}.\,} Hence one can say that h "is compatible with the group structure". Older notations for the homomorphism h(x) may be xh or xh, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that h(x) becomes simply x h. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right. In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.

Words

This table shows the example usage of word lists for keywords extraction from the text above.

WordWord FrequencyNumber of ArticlesRelevance
h25545950.425
homomorphism61870.227
u8196840.166
g8437580.142

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