Holomorph of a group

In mathematics , especially in group theory , the holomorph of a group G is a certain group labeled with , which contains both the group G and its automorphism group , or at least copies of these two groups. The holomorph makes it possible to show the inversions of certain sentences about complete groups and characteristically simple groups . There are two versions, one as a semi-direct product and one as a permutation group . ${\displaystyle \mathrm {Hol} (G)}$

Hol (G) as a semi-direct product

If the automorphism group of the group is designated , one sets ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle G}$

${\displaystyle \mathrm {Hol} (G)=G\rtimes \mathrm {Aut} (G),}$

wherein the (external) semi direct product to the natural operation of on G belongs. Thus the Cartesian product of G and has as the underlying set and the group operation is through ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \mathrm {Aut} (G)}$

${\displaystyle (g,\alpha )(h,\beta )=(g\alpha (h),\alpha \beta )}$

Are defined.

Hol (G) as a permutation group

A group G operates (from the left) naturally on itself, more precisely on its underlying set, by means of multiplication from the left and multiplication from the right. The multiplication from the left belongs to homomorphism

${\displaystyle \lambda \colon G\rightarrow S_{G},\,g\mapsto \lambda _{g}\colon h\mapsto gh}$

from to , wherein the symmetrical group is equipped with the group linkage. The multiplication from the right belongs to homomorphism ${\displaystyle G}$ ${\displaystyle S_{G}}$ ${\displaystyle S_{G}}$ ${\displaystyle (f,g)\mapsto f\circ g\colon x\mapsto f(g(x))}$

${\displaystyle \rho \colon G\rightarrow S_{G},\,g\mapsto \rho _{g}\colon h\mapsto hg^{-1}.}$

In this second operation one has to invert g to get a left operation, that is, a homomorphism as we defined it. ${\displaystyle G\rightarrow S_{G}}$

These two homomorphisms are injective and therefore define isomorphisms from G to subgroups or from (as in Cayley's theorem ). For a given g , the permutation of G is often called “the translation from the left with g ”. ${\displaystyle \lambda (G)}$ ${\displaystyle \rho (G)}$ ${\displaystyle S_{G}}$ ${\displaystyle \lambda _{g}\colon x\mapsto gx}$

We now define as the subgroup of and generated by . It is easy to show that for an element ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle S_{G}}$ ${\displaystyle \sigma \in \mathrm {Aut} (G)}$

${\displaystyle (1)\qquad \sigma \circ \lambda _{g}\circ \sigma ^{-1}=\lambda _{\sigma (g)}}$

holds, that is, normalized . Generating there and together is a normal divisor in . Similarly, one can show that the normalizer of in is. ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle S_{G}}$

In addition, one has , because a translation that is an automorphism maps 1 to 1 . So the (internal) semi-direct product is made up of and . Then it follows from equation (1) that the mapping defines an isomorphism between the external semi-direct product (with the natural operation of on G) and the holomorph . The two definitions of given here lead to isomorphic groups. ${\displaystyle \lambda (G)\cap \mathrm {Aut} (G)=1}$ ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle (g,\sigma )\mapsto \lambda _{g}\circ \sigma }$ ${\displaystyle G\rtimes \mathrm {Aut} (G)}$ ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \mathrm {Hol} (G)}$

It is easy to show that the permutation group is also the subgroup in created by and . (Note , where denotes the internal automorphism .) ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \rho (G)}$ ${\displaystyle \mathrm {Aut} (G)}$ ${\displaystyle S_{G}}$ ${\displaystyle \rho _{g}=\lambda _{g^{-1}}\circ \gamma _{g}}$ ${\displaystyle \gamma _{g}}$ ${\displaystyle x\mapsto gxg^{-1}}$

Since it defines an isomorphism of G on , every automorphism on has the form for an automorphism of G. Then the above relation (1) shows: ${\displaystyle g\mapsto \lambda _{g}}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \lambda _{g}\mapsto \lambda _{\sigma (g)}}$ ${\displaystyle \sigma }$

• Every automorphism of is the restriction of an internal automorphism of . ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Hol} (G)}$

Since is isomorphic to G , we get: ${\displaystyle \lambda (G)}$

• Each group G may be in a group H are embedded such that each automorphism of G , the restriction of an inner automorphism of H is.

It also follows from this:

• A subgroup of is characteristic if and only if it is the normal subgroup in . ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Hol} (G)}$

Examples

${\displaystyle \mathrm {Hol} (\mathbb {Z} _{3})\cong S_{3}}$, where the symmetric group is third degree . ${\displaystyle S_{3}}$

${\displaystyle \mathrm {Hol} (\mathbb {Z} )\cong D_{\infty }}$, where is the infinite dihedral group . ${\displaystyle D_{\infty }}$

Applications of the holomorph

• A group is called complete if its center consists only of the neutral element and all its automorphisms are internal. One proves: If a complete subgroup G is the normal subgroup of a group H , then it is even a direct factor of H. Conversely, one can show that a group is complete if it is a direct factor in every group in which it is contained as normal subgroup is. For this one uses the fact that under the given conditions there is a direct factor of the holomorph . ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Hol} (G)}$

• A group is said to be characteristically simple if the only characteristic subgroups are the trivial subgroup and the group itself. It is easy to show that all minimal normal subdivisions of a group are characteristically simple. Conversely, one can show that every nontrivial, characteristically simple group can be embedded in another group in such a way that it is the minimal normal divisor there. Since G and are isomorphic, it suffices to show that minimal normal divisor is in the holomorph . This follows easily from the fact mentioned above that a subgroup of is characteristic if and only if it is the normal subgroup in . ${\displaystyle \lambda (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Hol} (G)}$ ${\displaystyle \lambda (G)}$ ${\displaystyle \mathrm {Hol} (G)}$

Historical remark

The English term holomorphic to denote the construction presented here was introduced by William Burnside in 1897 . However, it appears earlier with other authors.

In German-language literature, the term "holomorphism of a group" was also used earlier .

Individual evidence

1. ^ Wilhelm Specht : Group theory. Springer-Verlag (1956), Chapter 1.3.5: The holomorph of a group.
2. ^ Joseph J. Rotman: An Introduction to the Theory of Groups. 4th edition (1999), p. 15.
3. ^ Derek JS Robinson : A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 0-387-94461-3 , section 1.6, page 37: The Holomorph.
4. ↑ ^{a b} Rotman (1999), p. 164.
5. ^ WR Scott: Group Theory. Dover Books on Mathematics, (1987), 2nd edition (1st edition 1964), ISBN 978-0-48665377-8 , p. 214.
6. ^ Derek JS Robinson: A Course in the Theory of Groups. Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , exercise 1.6.9.
7. ^ Wilhelm Specht: Group theory. Springer-Verlag (1956), ISBN 978-3-642-94668-4 , example 1 in section 1.3.6.
8. ↑ Scott (1987), p. 450 or Rotman (1999), p. 163.
9. ^ GA Miller, HF Blichfeldt, LE Dickson: Theory and Applications of Finite Groups. New York (1916), reprinted by Applewood Books (2012).
   W. Burnside: Theory of groups of finite order. 1st edition, Cambridge (1897), p. 228, online.
10. ↑ Andreas Speiser: The theory of groups of finite order (=  textbooks and monographs from the field of exact sciences - Mathematical series . No. 22 ). 4th edition. Birkhäuser Verlag, Basel / Stuttgart 1956, Chapter 9, § 40 Automorphisms of a Group , p. 121 .