First isomorphism theorem – Knowledge and References

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Group Theory

Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021

Gregory S. Chirikjian, Alexander B. Kyatkin

The preceding theorem is sometimes called the “fundamental theorem of homomorphisms” or the “first isomorphism theorem.” There are, in fact, a number of other isomorphism theorems which will not be discussed here.

The beta-transformation's companion map for Pisot or Salem numbers and their periodic orbits

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Journal Information

Published in Dynamical Systems, 2018

Bruno Melo Maia

Proposition 3.1 is related to a reformulation in our setting of a familiar algebraic result. Namely, is isomorphic, both as -algebras and as -modules, to . One defines the algebra homomorphism θ from to by sending x to β: this is clearly surjective, since is spanned as a -vector space by the powers of β. The kernel of θ is precisely <p(x) >, given that p(x) is the minimal polynomial of β. So, by the First Isomorphism Theorem for algebra homomorphisms, . This also shows that multiplication of an element of by x corresponds to multiplication of the corresponding element of by β. Finally, admits a -basis {1, x, …, xd − 1}, which defines an isomorphism between and . The pair becomes a -module if we define for , hence multiplication by β in corresponds to the linear map in (see [1, p. 478]).