Is every representation of $G$ over $\mathbb{K}$ trivial? True/False: Let G be a group with $|G| = p^n$ for a prime number $p$ and $n \in \mathbb{N}$, and let $\mathbb{K}$ be a field of characteristic $p$. 
Is every representation of $G$ over $\mathbb{K}$ trivial?
attempt:  False. Because we would need irreducibly.
Can someone please verify and provide some feedback? Thanks
 A: There are two things at stake here. $p$ is a prime number.
1 There exist  $G$ a $p$-group (i.e. $|G|=p^n$ with $n$ an integer), a field $\mathbb{K}$ of characteristic $p$ and a representation $\rho:G\rightarrow GL(n,\mathbb{K})$ such that $\rho$  is not the trivial representation. 
Hint :

 Take $\mathbb{K}:=\mathbb{Z}/p$ the field with $p$ elements. Find a non-trivial $p$-group $G$ in $SL(n,\mathbb{K})$ and justify that this will lead to a non-trivial group representation. 

Answer :

 Take $\mathbb{K}:=\mathbb{Z}/p$, $n:=2$, $G$ the subgroup of upper triangular matrices with $1$ 's on the diagonal in $SL(2,\mathbb{K})$ and $\rho:G\rightarrow SL(2,\mathbb{K})$ be the natural inclusion of $G$ into $SL(2,\mathbb{K})$. Clearly $\rho$ is not trivial (because $\rho(G)$ is of cardinal $p$) and $|G|=p$ so $G$ is a $p$-group. 

2 When we add the irreducibility then the representation is necessarily trivial see Irreducible Representation of a $p$-group over field of characteristic $p$ is trivial (Dummit and Foote 18.1 #22) .
