Internal semidirect product
This article describes a product notion for groups. See other related product notions for groups.
This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition with symbols
A group is termed an internal semidirect product of subgroups and if the following hold:
Note here that acts as automorphisms on by the conjugation action.
Equivalence with external semidirect product
Further information: Equivalence of internal and external semidirect product
Suppose is an internal semidirect product with normal subgroup and as the other subgroup. If we start out with and as abstract groups, and with the action of on (abstractly) which comes from the conjugation in , then the external semidirect product formed from these is isomorphic to .
- A subgroup which occurs as the normal subgroup for an internal semidirect product is termed a complemented normal subgroup, sometimes also called split normal subgroup.
- A subgroup which occurs as the permutable complement to a normal subgroup, is termed a retract. This is because there is a retraction from the whole group, to this subgroup, whose kernel is the normal subgroup.
- Every group is the internal semidirect product of itself and the trivial subgroup. In fact, it is an internal direct product of itself and the trivial subgroup.
- Given two groups and , is the internal semidirect product of and . In fact, it is the internal direct product.
- The symmetric group on any finite set of size at least two is the internal semidirect product of the alternating group and the two-element subgroup generated by any transposition. For instance, the symmetric group of degree three is the internal semidirect product of the subgroups and .
- The symmetric group of degree four is an internal semidirect product of the normal subgroup and a six-element subgroup (isomorphic to symmetric group of degree three) comprising the permutations that fix .
- The dihedral group of degree and order is the internal semidirect product of a cyclic subgroup of order (the rotations) and a cyclic subgroup of order (generated by a reflection).
- The cyclic group of order four has a normal subgroup of order two, but this normal subgroup has no complement.
Relation with other properties
Stronger product notions
Weaker product notions
Related subgroup properties
- Complemented normal subgroup is a normal subgroup having a permutable complement, and hence, part of a semidirect product.
- Retract is a subgroup having a normal complement, and hence, part of a semidirect product.
Related group properties
- Splitting-simple group is a group that cannot be expressed as an internal semidirect product of nontrivial subgroups.
- Complement to normal subgroup is isomorphic to quotient: In particular, given the normal subgroup for a semidirect product, we know the isomorphism type of the complement.
- Complements to normal subgroup need not be automorphic: Given a normal subgroup of a group with two complements and , it is not necessary that there exist an isomorphism of sending to . It is also not necessary that the actions of and on are the same.
- Complements to abelian normal subgroup are automorphic
- Complements to normal Hall subgroup are conjugate: This is the conjugacy part of the result commonly known as the Schur-Zassenhaus theorem.