Let F be a field and G group so the group algebra is defined as:
$F[G] = {\sum_{g\in G} c_{g} g: c_{g} \in F, g\in G}$
take the element $a= \sum_{g \in G} g$, clearly $a\in G$. Can I say that a is in the center of $F[G]$, i.e a comutes with every element in $F[G]$
Yes. $\sum_{g \in G}g$ is a central element of $F[G]$ assuming that $G$ is finite and this sum is well-defined.
Let $\sum_{g \in G} a_g g \in F[G]$, then $(\sum_{h \in G} h)(\sum_{g \in G} a_g g)=\sum_{g \in G}a_g (\sum_{h \in G}hg)$. Now the map $G \to G, h \mapsto hg$ is a bijection, as is the map $G \to G, h \mapsto gh$, so we have $\sum_{h \in G}hg=\sum_{h \in G} gh$. Thus $\sum_{g \in G}a_g(\sum_{h \in G} hg)=\sum_{g \in G}a_g(\sum_{h \in G} gh)=(\sum_{g \in G}a_g g)(\sum_{h \in G} h)$. This shows that $\sum_{h \in G}h$ is central.