Locoism said:The Attempt at a Solution
G/Z(G) = gnZ(G) for some g ε G and for any n ε N
choose some g-1 such that
g(g-1h) = g(hg-1)
and the same can be done switching the g and g-1
This doesn't feel right at all...
Locoism said:Homework Statement
Let G be a group. Show that G/Z(G) [itex]\cong[/itex] Inn(G)
The Attempt at a Solution
G/Z(G) = gnZ(G) for some g ε G and for any n ε N
choose some g-1 such that
g(g-1h) = g(hg-1)
and the same can be done switching the g and g-1
This doesn't feel right at all...
A group is a mathematical structure that consists of a set of elements and a binary operation that combines any two elements in the set to produce a third element. Groups are important in many areas of mathematics, including algebra, geometry, and physics.
An inner automorphism is a type of automorphism, which is a bijective mapping from a mathematical structure to itself. In the context of groups, an inner automorphism is a mapping that takes an element in the group and maps it to another element in the group by conjugation. In simpler terms, it is a transformation of a group that preserves its structure and operations.
Groups and inner automorphisms are closely related because inner automorphisms form a subgroup of the larger group. This subgroup is known as the inner automorphism group, and it consists of all the automorphisms that can be obtained by conjugating elements of the group. In other words, the inner automorphism group is a subgroup of the group that preserves its structure.
Groups and inner automorphisms are essential tools in mathematics and have many practical applications. In algebra, they are used to study symmetry and properties of equations. In geometry, they are used to understand the structure of shapes and objects. In physics, they are used to describe the symmetries and transformations of physical systems.
One example of a group and its inner automorphism group is the group of rotations of a regular hexagon. The group consists of six elements, which are the different ways the hexagon can be rotated. The inner automorphism group consists of three elements, which are the rotations by 120, 240, and 360 degrees. These rotations preserve the structure of the hexagon and can be obtained by conjugating elements of the group.