Automorphism group is transitive on non-identity elements implies aperiodic or prime exponent

From Groupprops
Jump to: navigation, search
The statement of this article contains an assertion of the form that for a certain kind of group, either every element has finite order (i.e., the group is a Periodic group (?)) or every non-identity element has infinite order (i.e., the group is a Torsion-free group (?) or aperiodic group). The actual statement in this case may be considerably stronger.
View other such statements

Statement

Suppose G is a nontrivial group that is a Group whose automorphism group is transitive on non-identity elements (?). In other words, for any two non-identity elements g,h \in G, there is an automorphism \sigma of G (i.e., an element of the automorphism group \operatorname{Aut}(G)) such that \sigma(g) = h.

Then, either G is a Torsion-free group (?) (i.e., all the elements of G have infinite order) or G is a Group of prime exponent (?) (i.e., there exists a prime number p such that every non-identity element of G has order p).

(For an abelian group, the condition that the automorphism group is transitive on non-identity elements is equivalent to the condition that the group is the additive group of a field. In particular, it is either an elementary abelian group or a vector space over the field of rational numbers. See additive group of a field for more.)

Related facts