Alternating group implies every element is automorphic to its inverse
From Groupprops
Contents
Statement
Let be any natural number. Let
denote the Alternating group (?) of degree
, i.e., the group of even permutations on a set of size
. Then,
is a Group in which every element is automorphic to its inverse (?). In other words, for every element
, there is an automorphism of
that conjugates
to
.
Related facts
Related facts about alternating groups
- Alternating group implies any two elements generating the same cyclic subgroup are automorphic
- Classification of ambivalent alternating groups:
is ambivalent for precisely the values
.
- Classification of alternating groups having a class-inverting automorphism:
has a class-inverting automorphism for precisely the values
.
Related facts about every element being automorphic to its inverse
- General linear group implies every element is automorphic to its inverse
- Projective general linear group implies every element is automorphic to its inverse
- Special linear group implies every element is automorphic to its inverse
- Projective special linear group implies every element is automorphic to its inverse
- Every element is automorphic to its inverse is characteristic subgroup-closed
- Normal subgroup of ambivalent group implies every element is automorphic to its inverse
Facts used
Proof
Consider the group , in which
is a normal subgroup of index two. Since
is a rational group, the elements
and
are conjugate. Thus, there exists
such that
.
Since is a normal subgroup of
, conjugation by
restricts to an automorphism of
.