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Factor groups

  1. Oct 27, 2011 #1
    Factor groups!!

    Please I just want to ask about factor groups.. how could a factor group G/A acts on A by conjugation, knowing that A is a normal & abelian subgroup of G..

    and what do we mean when we say that an element in a group has jus 2 conjugates??

    thanks in advance :)
     
  2. jcsd
  3. Oct 27, 2011 #2

    Deveno

    User Avatar
    Science Advisor

    Re: Factor groups!!

    the only possible way that makes sense is to define:

    (gA)(a) = gag-1. note that gag-1 is in A, since A is normal.

    however, we have to check that:

    if g'A = gA, then g'ag'-1 = gag-1

    (that is, that our action is well-defined).

    now if g'A = gA, g' = ga' for some a' in A. thus:

    g'ag'-1 = (ga')a(ga')-1

    = ga'aa'-1g-1

    but since A is assumed abelian, aa'-1 = a'-1a, so:

    ga'aa'-1g-1 = ga'a'-1ag-1

    = geag-1 = gag-1, which was to be proven.

    *******

    to say an element x has just two conjugates, means that gxg-1 has just two possible values (one of which is obviously x), no matter what g is.

    for example, consider the group Q = {1,-1,i,-i,j,-j,k,-k} under quaternial multiplication.

    (i2 = j2 = k2 = -1, ij = k, jk = i, ki = j, and
    ji = -k, kj = -i, ik = -j)

    (1)(i)(1) = i
    (-1)(i)(-1) = i
    (i)(i)(-i) = i
    (-i)(i)(i) = i
    (j)(i)(-j) = (-k)(-j) = kj = -i
    (-j)(i)(j) = (-j)k = -i
    (k)(i)(-k) = j(-k) = -i
    (-k)(i)(k) = (-k)(-j) = kj = -i

    the only conjugates i has is i, and -i.
     
    Last edited: Oct 27, 2011
  4. Oct 27, 2011 #3
    Re: Factor groups!!

    Thank you very much for your clear answer, it was really very useful for me :)
     
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