Group cohomology

Imagine you have a group of friends who always play together. You all have different strengths and weaknesses, but together you make a good team. Now, let's say you want to figure out how well your team works together. This is kind of like what group cohomology does, but instead of friends playing together, we have mathematical groups working together.

A group is just a collection of mathematical objects that follow some rules. The rules are kind of like the rules you and your friends follow when you play games together. For example, in a group, you might have a rule that says you can't play the same game twice in a row. Groups can do lots of things, like flip things over, rotate things, or even just keep things the same.

Now, when groups work together, they can do different things as a team than they can do on their own. This is where group cohomology comes in. Co what? Cohomology is just a fancy way of saying that we're looking at how things fit together. In a group, things fit together in different ways depending on how the group behaves (i.e. how it follows its rules).

Group cohomology helps us understand these different ways of fitting together. For example, let's say you have two groups that want to work together. They might have some trouble at first because they follow different rules. Group cohomology helps us figure out how to fit the two groups together so that they can work as a team. It's kind of like fitting puzzle pieces together!

Group cohomology can also help us measure how well a group works together. Just like how you might measure how well your team did in a game, group cohomology can measure how well a mathematical group worked together.

So, in summary, group cohomology is all about understanding how groups work together and how well they do it. We look at how they fit together and measure how well they work as a team. And just like playing games with friends, working with groups can be a lot of fun!