]=S_m (why?). Since kerf sits in H (why?), we can consider [H:kerf]. Since |H|=p, it follows that either [H:kerf]=1 or [H:kerf]=p (why?). If it's the former, we're done (why?). So suppose that [H:kerf]=p and deduce a contradiction.The non simplicity of a group refers to the fact that a group is not simple, which means it is not the most basic or fundamental form of the group. In other words, a non simple group has subgroups and is not considered to be the most basic form of a group.
The non simplicity of a group is determined by analyzing its subgroups. If a group has proper nontrivial subgroups, then it is considered to be non simple. This means that the group can be broken down into smaller subgroups, which makes it a non simple group.
A non simple group is significant because it allows for the study of groups at a more complex level. By understanding the structure and properties of non simple groups, we can better understand other groups and mathematical structures. Non simple groups also have applications in various fields such as physics, chemistry, and computer science.
No, a non simple group cannot be made into a simple group. This is because the non simplicity of a group is a fundamental characteristic of the group itself. It is not possible to change the structure or properties of a group to make it simple.
Some examples of non simple groups include the symmetric group, the alternating group, and the dihedral group. These groups have proper nontrivial subgroups, making them non simple. Other examples include the general linear group, the orthogonal group, and the special unitary group.