SO(2) is identified as the only simple group that is not semisimple, as stated in Mirman's book on group theory. This distinction arises because simple groups are typically defined as non-abelian and connected, yet SO(2) possesses a non-trivial center, which complicates its classification. The Wikipedia article on Simple Lie Groups highlights the lack of a universally accepted definition, leading to variations in understanding among authors regarding the characteristics of simple Lie groups.
PREREQUISITESMathematicians, students of group theory, and anyone interested in the classification of Lie groups and their properties.
Simple Lie groups
Unfortunately, there is no generally accepted definition of a simple Lie group. In particular, it is not necessarily defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether R is a simple Lie group.
The most common definition implies that simple Lie groups must be connected, and non-abelian, but are allowed to have a non-trivial center.