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Splittable group

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A group generated by proper subgroups and with normal in and (so that the quotient group is isomorphic to , cf. Normal subgroup). is called a split extension of the group by the group , or a semi-direct product of and . If the subgroups and commute elementwise, i.e. for all , , their semi-direct product coincides with the direct product . A semi-direct product of a group and a group is given by a homomorphism of into the group of automorphisms of . In this case, the formula

for all , , defines the multiplication in . In the case when and is the identity mapping, is called the holomorph of (cf. Holomorph of a group).

References

[1] D. Gorenstein, "Finite groups" , Chelsea, reprint (1980)


Comments

Conversely, if is a semi-direct product, then conjugation with in defines a homomorphism from which can be reconstructed, i.e.

As a set the semi-direct product of and is . The subsets , are subgroups that identify with and .

How to Cite This Entry:
Splittable group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Splittable_group&oldid=35137
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article