Lower central series

Imagine you have a big toy box that has lots of smaller boxes inside it, like Russian dolls. Each box inside the toy box is smaller than the one before it. The lower central series of a group is kind of like these smaller boxes.

When we talk about the lower central series of a group, we're looking at smaller and smaller pieces of the group that fit inside each other. Just like how the smaller boxes fit nicely inside the bigger ones. We start with the whole group as the biggest box, and then look at smaller and smaller subgroups contained in the bigger one.

To be a little more specific, we start by looking at the group itself (the biggest toy box). Then, we look at the group generated by all of the commutators of the group - this is the first "smaller box" that fits inside the group. Commutators are like little toys that measure how much two elements of the group "don't get along" when we try to swap them.

After we've found the group generated by all the commutators, we can look at the commutators in that group and make a new "smaller box" inside the first one. We keep doing this over and over until we've gotten to a point where we can't make any new groups that fit inside the last one we found.

The final result is a series of nested subgroups of the original group, each containing the one before it. These nested subgroups are the "smaller and smaller boxes" that we were talking about earlier. We call this series the lower central series of the group!