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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.
That is, an element u of a ring R is a unit if there exists v in R such that = =, where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. [1] [2] The set of units of R forms a group R × under multiplication, called the group of units or unit group of R.
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.
Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group. [13] In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL 2 ( F ) is non-abelian and has only one subgroup of order 2, so this 2-Sylow ...
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 'size' of the units is measured in general by a determinant called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when n is large. The torsion in the group of units is the set of all roots of unity of K, which form a finite cyclic group.
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.
This gives the circle group the structure of a one-parameter group, an instance of a Lie group. In fact, up to isomorphism, it is the unique 1-dimensional compact , connected Lie group. Moreover, every n {\displaystyle n} -dimensional compact, connected, abelian Lie group is isomorphic to T n {\displaystyle \mathbb {T} ^{n}} .