Difference between revisions of "Number of conjugacy classes in a subgroup may be more than in the whole group"

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(Created page with "==Statement== It is possible to have a finite group <math>G</math> and a subgroup <math>H</math> of <math>G</math> such that the [[fact about::number of conjugacy classe...")
 
 
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It is possible to have a [[finite group]] <math>G</math> and a [[subgroup]] <math>H</math> of <math>G</math> such that the [[fact about::number of conjugacy classes]] in <math>H</math> is more than in <math>G</math>.
 
It is possible to have a [[finite group]] <math>G</math> and a [[subgroup]] <math>H</math> of <math>G</math> such that the [[fact about::number of conjugacy classes]] in <math>H</math> is more than in <math>G</math>.
  
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==Related facts==
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===Similar facts===
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* [[Commuting fraction in subgroup is at least as much as in whole group]]: This says that the number of conjugacy classes in the subgroup is ''at least'' as much as the number of conjugacy classes in the whole group divided by the index of the subgroup.
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===Opposite facts===
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* [[Number of conjugacy classes in a subgroup of finite index is bounded by index times number of conjugacy classes in the whole group]]
 
==Proof==
 
==Proof==
  

Latest revision as of 05:00, 22 June 2012

Statement

It is possible to have a finite group G and a subgroup H of G such that the Number of conjugacy classes (?) in H is more than in G.

Related facts

Similar facts

Opposite facts

Proof

Example of the dihedral group of degree five

Further information: dihedral group:D10, cyclic group:Z5

The smallest pair of examples is where the group G is D_{10}, the dihedral group of degree five and order ten, and H is the subgroup is cyclic group:Z5. The group D_{10} has four conjugacy classes: one of involutions, one identity element, and two conjugacy classes in the cyclic subgroup of order five. On the other hand, H is an abelian group of order five hence has five conjugacy classes.

Other dihedral examples

Further information: element structure of dihedral groups

More generally, for odd n, the dihedral group of order 2n and degree n has (n + 3)/2 conjugacy classes, and the cyclic subgroup of order n has n conjugacy classes. For n \ge 5, the cyclic subgroup has more conjugacy classes than the whole group.

For even n, the dihedral group of order 2n has (n + 6)/2 conjugacy classes and the cyclic subgroup of order n has n conjugacy classes. For n \ge 8, the cyclic subgroup has more conjugacy classes than the whole group. The first example of this is cyclic group:Z8 in dihedral group:D16.