Finding Characteristic of a field , order of elements in Subgroup, Centre of Group 
Let $K$ be a field, $n$ a positive integer and $G$ a finite subgroup of $GL_n(K)$ such that $|G|>1$. Further assume that every $g\in G$ is upper triangular and all the diagonal entries of $g$ are $1$.
(A) Show that $\operatorname{char} K>0$.
(B) Show that the order of $g$ is a power of $\operatorname{char}K $, for every $g\in G$.
(C) Show that the centre of $G$ has at least two elements.

I have been following Joseph Gallian's Abstract Algebra and this question was asked in my End Term Exam last week and I couldn't solve any part. I tried it again today but failed to make any progress.
Can you please help me with it.
I understand the definitions and propositions, but I am completely struck on this.
Characteristic of $K$ is the least $n \in \mathbb{N}$ such that $n .1_K =0$, Center of $G = \{x \in G \mid gx =xg \text{ for all } g\in G\}$.
 A: For (a) and (b)
really it comes down to writing $g\in G$ as $(I+N)$ where $N$ is strictly upper triangular (and hence nilpotent). There's nothing to do when $g=e$ (i.e. $ N= \mathbf 0$ case), so consider $g \in G-\big\{e\big\}$.
Then $N^r= \mathbf 0$ for some minimal $r\in\big\{2,...,n\big\}$ -- i.e. $N^{r-1}\neq \mathbf 0 = N^r$. And since your group is of finite order, $g^k =e$ for some $k \geq 2$. But this means
$I = (I + N)^k = I + \binom{k}{1}N^1+ \binom{k}{2}N^2+....+ \binom{k}{k-1}N^{k-1} + N^k$
$\implies \mathbf 0 = \binom{k}{1}N^1+ \binom{k}{2}N^2+....+ \binom{k}{k-1}N^{k-1} + N^k$
which is a contradiction if $\text{char }\mathbb F = 0$
Why? Consider multiplying each side by $N^{r-2}$ to get $k\cdot N^{r-1} = \mathbf 0$
Thus $\text{char }\mathbb F = p$, for some prime $p$
This also tells you that $k\%p =0$ and gives you part (b).
For (c)
You can write $\big \vert G\big \vert = p^m \cdot z$.  Where $z\%p \neq 0$. If $z\neq 1$ then applying Cauchy's Theorem gives the existence of an element of an order that isn't a multiple of $p$-- contradicting (b).  Hence $G$ is a $p$ group and must have a center with more than one element.
