In the group ring $\mathbb{Z}_p [G]$, what elements satisfy $(\sum a_g g)^p = \sum a_g g^p$? Here $\mathbb{Z}_p$ is the ring of integers in $\mathbb{Q}_p$.
Preferably I would want to know this for a general group $G$, but I have been concentrating on the case $G = (\mathbb{Z} / p^n \mathbb{Z})^{\times}$ as a starting point.
I have also looked at the case when $\sum a_g g$ has finite order and I deduced that this implies $\sum (a_g g)^p = \sum a_g g^p$ (I wrote $\sum a_g g$ as $\sum p^i a_{g,i} g$ so we now have $1 = (\sum p^i a_{g,i} g)^k$. By equating coefficients we see that $\sum a_{g,0} g$ is both a unit and a zero-divisor unless $\sum a_{g,i} g = 0$ for all $i>0$. Finally we just equate coefficients of $(\sum a_g g)^p = \sum a_g g^p$.)
Any help on the general case or special case would be greatly appreciated.
 A: For a finite Abelian group $G$ you can use the linear characters of $G$ ( and an extension $\mathbb{Z}_{p}[\omega]$ for a suitable root of unity) to translate the problem as follows: there are $|G|$ different homomorphisms from $G$  to $\mathbb{Z}_{p}[\omega]^{\times},$ (the linear characters of $G$) and your equation is satisfied in $G$ if and only if $\left( \sum_{g} a_{g} \lambda (g)\right)^{p} = \sum_{g}a_{g} \lambda(g^{p})$ for every choice of such a linear character $\lambda.$ 
For a general finite group $G,$ use of the augmentation ideal of the group ring gives the necessary condition that $\sum_{g}a_{g}$ is one of the $p$ solutions of $x^{p} = x$ in $\mathbb{Z}_{p}.$
A: If $G$ has no $p$-torsion, then the Jacobian matrix of this system of equations is invertible modulo $p$ (the derivative of the left side vanishes mod $p$, and the derivative of the right side is a permutation matrix). Hence each mod $p$ solution lifts to a unique solution over $\mathbb Z_p$.
If in addition $G$ is abelian, then every function mod $p$ is a solution. 
We can calculate the solutions in this case using Geoff Robinson's answer. Because in this case composing with the $p$th power is a permutation with the character, the Fourier coefficient associated to each character is a $q-1$ root of unity or zero, where $q$ is the order of the finite field over which that character, modulo $p$, is defined.  So we can calculate the lift by taking an arbitrary mod $p$ function on $G$, applying the Fourier transform, obtaining a function on $\hat{G}$, taking Techmuller lifts, and applying the inverse Fourier transform.
EDIT : I think my first sentence is actually only true when $G$ is abelian and $p$-torsion free. We can write the derivative as a sum of $p$ terms, which are identical if $g$ is commutative.
