Solvable group

A solvable group is a group that can be "unwrapped" like a present. Imagine you have a box with different layers of wrapping paper around it. Each layer represents a subgroup of the group. A group is solvable if you can remove each layer of wrapping paper to reveal a simpler subgroup until you get to a simple group (a group that cannot be further "unwrapped").

For example, let's say we have a group of people who want to form a dance group. They start out as one big group, but then they split into smaller groups based on their dance preferences. These smaller groups can then split further into even smaller groups based on other preferences until eventually, we have groups of people who all like the same type of dancing. These groups are simple because they cannot be split any further.

In maths, we call each layer of wrapping paper a normal subgroup and a solvable group is a group where we can create a series of normal subgroups, each containing the previous one, until we get to the simple group. So, just like unwrapping a present, we can keep removing each layer of normal subgroup until we reach the simple group at the center.

In summary, a solvable group is a group that can be "unwrapped" by removing normal subgroups until we get to a simple group, just like unwrapping a present by removing each layer of wrapping paper to reveal the gift inside.