Classification of ambivalent alternating groups
This article classifies the members in a particular group family alternating group that satisfy the group property ambivalent group.
Contents
Statement
The alternating group is an ambivalent group for precisely the following choices of
:
.
Note that in the proof, we show that in each of these cases, for every element in the alternating group, there is an element of order two in the alternating group conjugating it to its inverse. Thus, we show that the set of for which
is strongly ambivalent is precisely the same:
.
Related facts
Related facts about alternating groups
- Classification of alternating groups having a class-inverting automorphism: This turns out to be
.
- Alternating group implies every element is automorphic to its inverse
- Finitary alternating group on infinite set is ambivalent
Related facts about symmetric groups
- Symmetric groups are rational
- Symmetric groups are rational-representation
- Symmetric groups are ambivalent
General information pages
Facts used
Proof
By fact (1), a product of cycles of distinct odd lengths is conjugate to its inverse if and only if
is even. Equivalently, it is conjugate to its inverse if and only if the number of
s that are congruent to
modulo
is even. Note also that if it is conjugate to its inverse, we can choose as our conjugating element an element of order two: the product of transpositions described above.
What this boils down to for 
Thus, the problem reduces to the following: for what can we write
in such a way that all
are distinct, and the number of
that are congruent to
modulo
is odd? These are precisely the
for which
is not ambivalent.
We quickly see the following:
-
can be written in this form, because we can take
.
-
can be written in this form, because we can take
.
-
can be written in this form, because we can take
.
-
can be written in this form, because we can take
.
The only cases left are , and it is readily seen that a decomposition into
of the above form is not possible for these
.