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