Explore BrainMass
Share

Group Action, Conjugates and Conjugation

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

Consider the group action of on itself via conjugation.
={ }, and
a) Find all the elements of that are fixed by the element r.
b) Let G be a group, and consider the action of G on itself via conjugation. Let g to G. Prove or disprove that the set of all elements of G that are fixed by g is a subgroup of G.
c) Find all the elements of that fix the element s.
d) Let G be a group, and consider the action of G on itself via conjugation. Let g G. Prove or disprove that the set of all elements of G that fix g is a subgroup of G.

Please see the attached file for the fully formatted problems.

© BrainMass Inc. brainmass.com October 24, 2018, 8:28 pm ad1c9bdddf
https://brainmass.com/math/group-theory/group-action-conjugates-conjugation-98075

Attachments

Solution Preview

(a) All the elements in D14 that are fixed by r is H=<r>={1,r,r^2,...,r^6}.
Actually, from the result of (b), we know that H is a subgroup of D14.
Each element in <r> commutes with r and thus is fixed by r. s does not
fix r and <r> is the greatest real subgroup of D14. Thus H=<r>.
(b) The statement is true.
Proof: Suppose H is the ...

Solution Summary

Group Action, Conjugates and Conjugation are investigated.

$2.19
See Also This Related BrainMass Solution

Conjugacy classes: symmetric and alternating groups

The conjugacy classes in Sn correspond to partitions of n, and are determined by cycle structure.

Which conjugacy classes of Sn that are in An split into two conjugacy classes in An? Please give a necessary and sufficient condition.

Sn is the symmetric group of degree n, and An is the alternating group of degree n.

View Full Posting Details