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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.
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.
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.
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 ...
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 ...
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.
In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings have exactly the same properties. If R and S are rngs, then the corresponding notion is that of a rng homomorphism, [a] defined as above except without the third condition f(1 R) = 1 S. A rng ...
Using 1 to denote the multiplicative identity of the ring R, and denoting the group unit by 1 G, the ring R[G] contains a subring isomorphic to R, and its group of invertible elements contains a subgroup isomorphic to G. For considering the indicator function of {1 G}, which is the vector f defined by