Welcome to Group Theory!
Group Theory is a vibrant, wide area of current research in mathematics, computer science and mathematical/theoretical physics. There are many applications of group theory to the study of geometric objects, to topology and in many cases their links to other branches of algebra are quite well understood.
So, why study group theory?
You might study it because you've got a research question that somehow involves symmetry --- constraint problems in computer science can be solved more efficiently when a little's known about the solution space. You might need to make some difficult calculations in a complicated topological space which become easier by passing into structures called their fundamental groups.
You might even want to make millions as a cryptographer --- current methods in cryptography are hard to crack, but become very easy when hacking with a quantum computer. There are good reasons to hope that groups hold the key to codes that can't be broken easily by quantum computers, just in case someone invents one and fancies stealing your identity. We've just not come up with the perfect code yet.
Perhaps you just want to study it because it's fun and not really difficult to get into. Group theory can be understood at a moderate level by high-school level students, and in fact well enough by interested undergraduate students for them to produce original research. This is in stark contrast to the New Math introduced in American high schools in the mid-20th century, which consisted of teaching young teenagers abstract subjects like Set Theory and in certain cases Category Theory, both of which are very difficult to get an idea about without a solid grounding in undergraduate mathematics.
It's also a very beautiful subject. Many familiar mathematical objects are some sort of group - some in more than one way. Understanding a little about groups can make it easier to understand these objects too.
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