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Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n.
For that group all permutations of the three non-identity elements are automorphisms, so the automorphism group is isomorphic to (which itself is isomorphic to ). In Z p {\displaystyle \mathbb {Z} _{p}} for a prime number p , {\displaystyle p,} one non-identity element can be replaced by any other, with corresponding changes in the other elements.
The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1. In the simple case n = 1, the group U(1) corresponds to the circle group, isomorphic to the set of all complex numbers that have absolute value 1, under multiplication ...
The full automorphism group of Q 8 is isomorphic to S 4, the symmetric group on four letters (see Matrix representations below), and the outer automorphism group of Q 8 is thus S 4 /V, which is isomorphic to S 3. The quaternion group Q 8 has five conjugacy classes, {}, {¯}, {, ¯}, {, ¯}, {, ¯}, and so five irreducible representations over ...
More generally, for the ring of integers R in a number field F, Dirichlet's unit theorem states that R × is isomorphic to the group where is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is = +, where , are the number of real embeddings and the number of pairs of complex embeddings of F, respectively.
This is a bijection if and only if r is coprime with n, so the automorphism group of Z/nZ is isomorphic to the unit group (Z/nZ) ×. [18] Similarly, the endomorphism ring of the additive group of Z is isomorphic to the ring Z. Its automorphism group is isomorphic to the group of units of the ring Z, which is ({−1, +1}, ×) ≅ C 2.
The set of all nonzero scalar matrices forms a subgroup of GL(n, F) isomorphic to F ×. This group is the center of GL(n, F). In particular, it is a normal, abelian subgroup. The center of SL(n, F) is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of nth roots of unity in the field F.
The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition. In mathematics , an isomorphism is a structure-preserving mapping (a morphism ) between two structures of the same type that can be reversed by an inverse mapping .