Detail in proof that groups of order $mp$ with $1The following is from Aluffi's Algebra: 

In the underlined equality, where $1$ and $p-1$ comes from? If it uses The Class Formula (the picture below) then $|Z(G)|=1$ and why there are $p-1$ elements in $A$. 


You are assuming that there must exist $H_1, \dots, H_{p+1}$ subgroups of order $p$ of $G$ with $p-1$ elements of order $p$ each. Note that this accounts for a total of $(p+1)(p-1)$ elements of order $p$, becase no element of these can be in two of the subgroups at the same time (as $H_i \cap H_j = 1$ when $i \neq j$). We also have that $1 \in G$, so this gives a lower bound of $1 + (p-1)(p+1)$.
