Core of a subgroup
From Encyclopedia of Mathematics
Let be a subgroup of G. The core of H is the maximal subgroup of H that is a normal subgroup of G. It follows that \mathrm{core}_G (H) = \bigcap_g H^g \ ,\ \ \ H^g = gHg^{-1} If the index [G:H] = n < \infty, then [G:\mathrm{core}_G (H)] divides n!.
Let g(xH) = (gx)H and define the permutation representation of G on the set of right cosets of H in G (cf Coset in a group). Then its kernel is the core of H in G.
References
[a1] | M. Suzuki, "Group theory" , I , Springer (1982) |
[a2] | W.R. Scott, "Group theory" , Dover, reprint (1987) (Original: Prentice-Hall, 1964) |
How to Cite This Entry:
Core of a subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Core_of_a_subgroup&oldid=39865
Core of a subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Core_of_a_subgroup&oldid=39865
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article