For any $k$ sums $T_k = 1/|G|\sum_{g\in G} g^k$ belongs to the center of the group algebra.
For $k=2$ they "are" (up to details and interpretation) [Frobenius-Schur indicators][1]. 
For $k>2$ similar things called "higher" FS indicators.

**Question 1**
As any element of center $T_k$ acts by scalars in any irreducible representation, for $k=2$
the scalar can be $-1,0,1$ and brings important group theory information on the type of irrep 
(see WP and/or  [this MO question][2]). 

Is it true that values of higher $T_k$ in irreps are also integers ? Does it have some group theory interpretation ?  (These MO questions [1][3], [2][4] discuss FS  and relate them to dim of invariants on tensor powers of representation, but it seems different from my question.)

**Question 2**

How far the linear space/ algebra generated by  $T_k$ is from the whole center of group algebra?
(For cyclic group Z/n for k>0 all $T_k = 0$, so quite far from the center, but what can be said in general and in particular for S_n,A_n, GL_n(F_q), UT(n,q)  )?


PS

**Bonus Question 3** WP- article contains the following remark: 

> It resembles the Casimir invariants for Lie algebra irreducible representations.

In the sense that Casimirs are the center of U(g), while T_k are in the center of C[G].

Question: are there some more precise analogies ? 

In some my research we studied the center of U(gl_n) and its loop algebra,  
so such expression resembles "Gelfand" generators of U(gl_n) (which are traces of matrices "E^k", "E" given [here][5]), but I do not see any way
to go beyond "resemble". While it would be nice seems now I am trying to think on some finite group questions.
  


  [1]: http://en.wikipedia.org/wiki/Frobenius%25E2%2580%2593Schur_indicator
  [2]: http://mathoverflow.net/questions/46900/are-there-real-vs-quaternionic-conjugacy-classes-in-finite-groups
  [3]: http://mathoverflow.net/questions/20795/frobenius-schur-indicator-and-irreducible-representations-over-r
  [4]: http://mathoverflow.net/questions/42653/number-of-n-th-roots-of-elements-in-a-finite-group-and-higher-frobenius-schur-ind
  [5]: http://en.wikipedia.org/wiki/Capelli_identity