Regular p-group
A regular p-group is a special kind of group where everything in the group can be reached by multiplying a specific element in the group called the generator, and its inverse, in different ways.
Think of it like a treasure hunt. You have a map that shows you where to start, and a set of instructions on how to find the treasure. In a regular p-group, the generator is like your starting point, and the instructions are all the different ways you can reach the different elements in the group.
But there's one important rule: you can only use the generator and its inverse to move around the map. This makes sure that you can only find the elements that are "reachable" from the starting point.
And just like how every treasure hunt has a different map and set of instructions, every regular p-group has its own generator and specific set of rules for how to reach all the elements in the group.
So in summary, a regular p-group is a special kind of group where everything can be reached by following a set of rules using a specific element called the generator and its inverse. It's like a treasure hunt with a map and instructions, but with math instead of treasure.
Think of it like a treasure hunt. You have a map that shows you where to start, and a set of instructions on how to find the treasure. In a regular p-group, the generator is like your starting point, and the instructions are all the different ways you can reach the different elements in the group.
But there's one important rule: you can only use the generator and its inverse to move around the map. This makes sure that you can only find the elements that are "reachable" from the starting point.
And just like how every treasure hunt has a different map and set of instructions, every regular p-group has its own generator and specific set of rules for how to reach all the elements in the group.
So in summary, a regular p-group is a special kind of group where everything can be reached by following a set of rules using a specific element called the generator and its inverse. It's like a treasure hunt with a map and instructions, but with math instead of treasure.