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Difference between revisions of "Core of a subgroup"

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Let $H$ be a subgroup of $G$. The core of $H$ is the maximal subgroup of $H$ that is normal in $G$ (cf. also [[Normal subgroup|Normal subgroup]]). It follows that
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Let $H$ 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^h \ ,\ \ \ H^g = gHg^{-1}
 
\mathrm{core}_G (H) = \bigcap_g H^h \ ,\ \ \ H^g = gHg^{-1}
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If the index $[G:H] = n < \infty$, then $[G:\mathrm{core}_G (H)]$ divides $n!$.
 
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 $G$. Then its kernel is the core of $H$ in $G$.
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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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''I''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.R. Scott,  "Group theory" , Dover, reprint  (1987)  (Original: Prentice-Hall, 1964)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''I''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.R. Scott,  "Group theory" , Dover, reprint  (1987)  (Original: Prentice-Hall, 1964)</TD></TR></table>
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[[Category:Group theory and generalizations]]

Revision as of 18:30, 13 October 2014

Let $H$ 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 $$ \mathrm{core}_G (H) = \bigcap_g H^h \ ,\ \ \ H^g = gHg^{-1} $$ 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=33614
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article