Namespaces
Variants
Actions

Difference between revisions of "Core of a subgroup"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(LaTeX)
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202401.png" /> be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202402.png" />. The core of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202403.png" /> is the maximal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202404.png" /> that is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202405.png" /> (cf. also [[Normal subgroup|Normal subgroup]]). It follows that
+
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
 +
$$
 +
\mathrm{core}_G (H) = \bigcap_g H^h \ ,\ \ \ H^g = gHg^{-1}
 +
$$
 +
If the index $[G:H] = n < \infty$, then $[G:\mathrm{core}_G (H)]$ divides $n!$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202406.png" /></td> </tr></table>
+
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 $G$.
 
 
If the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202407.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202408.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c1202409.png" />.
 
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c12024010.png" /> and define the permutation representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c12024011.png" /> on the set of right cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c12024012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c12024013.png" />. Then its kernel is the core of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c12024014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120240/c12024015.png" />.
 
  
 
====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>

Revision as of 18:28, 13 October 2014

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). 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 $G$. 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