Not an algebraist myself, but this is interesting. Few things I do know:
Read ------------------------------- Quickly think of
algebraically closed field --------- C $\mathbb C$
abelian group ---------------------- (R;+)$(\mathbb R;+)$
non-abelian group ----------------- (SO(n,R);x) $(\operatorname{SO}(n,\mathbb R);\times)$ ; (n>=2)$n>2$)
simple group ----------------------- (An;o) $(A_n; \circ)$ ; (n>=5)$n\ge5$)

