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Unformatted text preview: 240 5. ACTIONS OF GROUPS Deﬁnition 5.1.15. Consider the action of a group G on its subgroups by
conjugation. The stabilizer of a subgroup H is called the normalizer of H
in G and denoted NG .H /.
According to Corollary 5.1.14, if G is ﬁnite, then the number of distinct subgroups xH x 1 for x 2 G is
ŒG W NG .H / D jG j
:
jNG .H /j Since (clearly) NG .H / Ã H , the number of such subgroups is no more
than ŒG W H .
Deﬁnition 5.1.16. Consider the action of a group G on itself by conjugation. The stabilizer of an element g 2 G is called the centralizer of g in
G and denoted Cent.g/, or when it is necessary to specify the group by
CentG .x/.
Again, according to the corollary the size of the conjugacy class of g ,
that is, of the orbit of g under conjugacy, is
ŒG W Cent.g/ D jG j
:
jCent.g/j Example 5.1.17. What is the size of each conjugacy class in the symmetric
group S4 ?
Recall that two elements of a symmetric group Sn are conjugate in Sn
precisely if they have the same cycle structure (i.e., if when written as a
product of disjoint cycles, they have the same number of cycles of each
length). Cycle structures are parameterized by partitions of n.
(a)
(b) (c) There is only one element of cycle structure 14 , namely, the identity.
There are six 2–cycles. We can compute this number by dividing
the size of the group, 24, by the size of the centralizer of any
particular 2cycle. The centralizer of the 2cycle .1; 2/.3/.4/ is
the set of permutations that leave invariant the sets f1; 2g and
f3; 4g, and the size of the centralizer is 4. So the number of 2–
cycles is 24=4 D 6.
There are three elements with cycle structure 22 . In fact, the
centralizer of .1; 2/.3; 4/ is the group of size 8 generated by
f.1; 2/; .3; 4/; .1; 3/.2; 4/g. Hence the number of elements with
cycle structure 22 is 24=8 D 3. ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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