Groups are important because they are the algebraic way to describe symmetry: if you have some operation that leaves a thing invariant (e.g. rotating an equilateral triangle so that a vertex lands on where another started), then the operation is invertible and the inverse leaves the thing invariant. You can compose such operations and the composition will still be invariant. The identity function always leaves everything invariant. So your symmetry operations form a group.

Slightly trickier is that every group is a set of symmetry operations for something. So groups exactly capture the idea of symmetry. To a mathematician, "group" and "symmetries" are synonymous.

Finite groups can be interesting as the symmetries of e.g. molecules (e.g. rotating atoms around onto each other), which can tell you something about molecular structure, energy levels, spectra, bonding potential, etc. Infinite groups appear in physics (e.g. the laws of physics are the same when you rotate or translate your coordinates by arbitrary amounts). Symmetry also comes up as a way to study other mathematical objects, and mathematicians might just want to know what all possible groups look like.

The presence of ann invariant implies invertibility! That doesn’t seem true.