G. Cutolo A note on central automorphisms of groups Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei
(9) Mat. Appl., 3, (1992),
pp. 103–106. MathSciNetZentralblattCommentsAbstractFull Text
Part of the main (and only) theorem in this paper can be extended to
group endomorphisms. I did it in my PhD thesis.
If G is a group and θ is an endomorphism
of G then θ is said to be centralizing if and only if
every element g of G commutes with its
image gθ. So, this condition is exactly the
requirement (i) in parts (a) and (b) of the theorem in [3].
Also, θ is central if [G,θ] is contained in
Z(G), while it is normal if θ
commutes with every inner automorphism of G (which amounts to
saying that [G,θ] centralises Gθ). These two properties are
equivalent if θ is an automorphism (or, at least, if it
is surjective) but there exist non-central normal endomorphisms in any
nonabelian group—for instance the zero endomorphism.
The result in the thesis is the following:
Let θ be a centralizing endomorphism of a
group G and assume that, for every element g
of G, the subgroup 〈[g,θ]〉 is normal
in G. Then:
θ is normal;
if g is an element of G such that either
[g,θ] has infinite order or g has odd order, then
[g,θ] lies in Z(G);
if θ is not central then [G,θ] has a G-invariant subgroup
isomorphic to the quaternion group of order 8.
Clearly, the fact that condition (c) of the theorem in [3] implies (a)
is a special case of this result.
Part of the main (and only) theorem in this paper can be extended to group endomorphisms. I did it in my PhD thesis.
If G is a group and θ is an endomorphism of G then θ is said to be centralizing if and only if every element g of G commutes with its image gθ. So, this condition is exactly the requirement (i) in parts (a) and (b) of the theorem in [3]. Also, θ is central if [G,θ] is contained in Z(G), while it is normal if θ commutes with every inner automorphism of G (which amounts to saying that [G,θ] centralises Gθ). These two properties are equivalent if θ is an automorphism (or, at least, if it is surjective) but there exist non-central normal endomorphisms in any nonabelian group—for instance the zero endomorphism. The result in the thesis is the following:
Let θ be a centralizing endomorphism of a group G and assume that, for every element g of G, the subgroup 〈[g,θ]〉 is normal in G. Then:
Clearly, the fact that condition (c) of the theorem in [3] implies (a) is a special case of this result.
Dismiss