Talk:Conjugacy class

Centralizer is a subgroup

Reverting to previous text , as the centralizer is indeed a subgroup of G; if a,b in C_G(g), then:

• C_G is not empty, since (g*g)*g = g*(g*g).
• b * g = g * b → b ^-1 * g = g * b ^-1; thus b ^-1 is in C_G(g)
• (a*b ^-1)*g = a*(b ^-1*g) = a*(g*b ^-1) = (a*g)*b ^-1 = (g*a)*b ^-1 = g*(a*b ^-1) → a*b ^-1 in C_G(g).

thus by elementary group theory, C_G(g) is a subgroup of G.

It is true that C_G(g) is not a normal subgroup of G; but that's not what was stated in the article. Maybe this should be added to the article.User:chas_zzz_brown

class equation

there should be written something about general (not only conjugacy) class equation; rather in group action than here, but here class equation redirects… konradek (talk) 15:29, 17 October 2008 (UTC)

Example in conjugacy class equation

I think this example could be spiced a little bit... Let's say, proving that every group of order p² is abelian is the next logical step, and quite trivial: the class equation for a hypothetical non-abelian group of order p² is |G| = |Z(G)| + |H|, where H includes all elements of order p that are not in the center (of order p) - but this is absurd, because p (p - 1) does not divide p² unless p = 2, and in this case there are only two groups of order 4 (both abelian). Albmont (talk) 16:49, 27 May 2009 (UTC)

(I don't know how to start a discussion/edit so here is my go:) I'm just a student but the sentence "Consider a finite p-group G (that is, a group with order p^n, where p is a prime number and n > 0). " is in contradiction with the definition of a p-group on wikipedia ( "p-group is a periodic group in which each element has a power of p as its order" )

so it's not |G| = p^n but for each g belongs to G, exists n belongs to N* such that |g| = p^n with p prime. — Preceding unsigned comment added by 67.212.8.144 (talk) 16:27, 6 December 2011 (UTC)

Examples

The symmetric group S4, consisting of all 24 permutations of four elements, has five conjugacy classes. And you can compute that there are five conjugacy classes by taking the integer partition of 4. But how can you compute how many elements each conjugacy class has? PJ Geest (talk) 14:56, 1 June 2009 (UTC)

Sometimes the function of the partition is called "n!/z". If the partition has a1 1s, a2 2s, a3 3s, and a4 4s, then the conjugacy class has size 4!/( a1! * 1^a1 * a2! * 2^a2 * a3! * 3^a3 * a4! * 4^a4 ). For instance the partition 4 = 1 + 1 + 2 corresponds to the conjugacy class containing the permutation [1243] = (1)(2)(3,4) whose centralizer is Sym({1,2}) x < (3,4) > of order 2! * 2^1, so the conjugacy class has size 4!/4 = 6 = 4!/( 2! * 1^2 * 1! * 2^1 * 0! * 3^0 * 0! * 4^0 ). In a larger symmetric group, the partition 7 = 2+2+3 corresponds to the conjugacy class containing [2143675]=(1,2)(3,4)(5,6,7) whose centralizer is < (1,2),(3,4),(1,3)(2,4) > x <(5,6,7)> = <(1,2)> wr Sym( { {1,2}, {3,4} } ) x <(5,6,7)> of order 2^2*2! * 3^1, and so the conjugacy class has size 7!/( 0!* 1^0 * 2! * 2^2 * 1! * 3^1 * 0! * 4^0 * 0! * 5^0 * 0! * 6^0 * 0! * 7^0 ) = 7!/( 2^2*2! * 3^1 ) = 7!/24 = 210. JackSchmidt (talk) 19:53, 1 June 2009 (UTC)

Error in class equation

Consider S_3 has 3 calsses C_1={e} , C_2={(acb),(bac),(cba)} , C_3 = {(cab),(bca)}. Center of S_3 is Z(S_3) = {e}. If we apply the equation exactly as it is written we'll get 6 = 1 + 6/1 + 6/3 + 6/2 and so 6=12 which is false.I think that sum should be a bit smaller. What do you think ? Stefan.petrea (talk) 17:00, 30 November 2009 (UTC)

If you exclude the conjugacy class of the identity element from that sum the identity becomes true(but that's not specified in the formula, is it so obvious ?) Stefan.petrea (talk) 17:06, 30 November 2009 (UTC)

I believe it is specified clearly and in a standard way. The sum is over the classes of size bigger than 1. The sum would be incorrect if it were over all non-identity classes; any abelian group gives a counterexample. JackSchmidt (talk) 17:41, 30 November 2009 (UTC)

Ok, then I suggest the formula be changed to $|G|=|Z(G)|+\sum _{i=1\land [G:H_{i}]>1}^{i=|S|}[G:H_{i}]$ , what do you think ? —Preceding unsigned comment added by Stefan.petrea (talk • contribs) 20:35, 30 November 2009 (UTC)

Section on motivation?

I think it would be helpful to have a section on the motivation of conjugation. Why is the concept useful? Why was it defined? Zaunlen (talk) 14:54, 10 November 2019 (UTC)

A complete answer to your question would need a complete history of the subject. However, I think that the problem is the style of the lead. I have edited it, and I hope that the new version provides a sufficient answer to your questions. D.Lazard (talk) 15:56, 10 November 2019 (UTC)

Thanks. I don't think a complete history is necessary. In order to show that the concept is useful, it suffices to name a few applications. Which text passage is it that now explains the usefulness of the concept? Zaunlen (talk) 19:20, 14 November 2019 (UTC)

Conjugacy of subgroups and general subsets

A005432(4) = 30 distinct subgroups (left) and A000638(4) = 11 conjugacy classes of subgroups (right) of S₄

I suppose that these diagrams could serve as an example for this section. The diagram on the right contains two C₂ vertices and two C₂² vertices, which illustrates the statement: "Conjugate subgroups are isomorphic, but isomorphic subgroups need not be conjugate." Watchduck (quack) 10:47, 19 November 2022 (UTC)