1) How is center of any group is the union of $1$-element conjugacy classes in the group ?
2) For a $p$-group the size of every conjugacy class is a power of $p$?
3) By using the above two properties how can i prove that A nontrivial $p$-group has a non-trivial center?.
Being a beginner,please guide me.
Any help is great.
1) If $g\in G$ is the center then it commutes with every other, i.e $\forall h\in G \quad hgh^{-1}=g$. Hence the conjugacy class of $g$ has a single element. (The other direction is just reversing the argument).
2)As each conjugacy class is a subgroup, then by Lagrange's theorem you have the result.
3) Using the class equation, if the group has order $p^k$, then:
$$ p^k=\sum_\text{classes C of size 1}1+\sum_\text{classes C of size > 1} |C| $$ By 1) the first sum is the order of the center, and by 2) the second sum is divisible by p, hence $|Z(G)|=p^k-\sum_\text{classes C of size > 1}|C|$ is divisible by p. Hence the result.