NonSimplicity of a Group of Order 36

Given a group $G$ whose order is 36, demonstrate why $G$ is not simple by using the extended Cayley theorem and considerations of cosets and homomorphisms.

In this problem, you will explore why a group $G$ of order 36 is not simple. A simple group has no nontrivial normal subgroups other than the group itself and the identity subgroup. This concept forms a critical cornerstone in group theory, especially when understanding the structure and classification of groups. By disproving the simplicity of $G$, you will delve into important tools such as the extended Cayley theorem and the idea of cosets and homomorphisms.

The order of the group, 36, is a composite number, offering a gateway to employing Sylow theorems. The Sylow theorems provide conditions under which a group possesses subgroups of a given prime power order. This plays a vital role in analyzing whether certain subgroups are normal, thus initially exploring the possibility of $G$ being simple.

Using Cayley’s theorem, you can represent a group $G$ as a subgroup of a symmetric group, which revolves around the concept of permutation representation. This representation helps us inspect the structures within $G$. Furthermore, consider how cosets help in breaking down the composition of groups in Lagrange’s theorem and how their role in forming quotient groups provides insights into the simplicity or complexity of a given group. Through the implications of homomorphisms, especially the properties of kernel and image, these form essential aspects in the examination of the group’s structure, eventually converging on the conclusion regarding the existence of a nontrivial normal subgroup in $G$. By drawing on these concepts, you can articulate and expose the complexity embedded within the group, revealing its non-simplicity.