📘 group theory

1. **Problem Statement:** Suppose $G$ is a finite group with exactly one nontrivial proper subgroup. We want to prove that $G$ is cyclic and that its order $|G| = p^2$ for some pri

1. **Problem Statement:** We consider the trivial representation of a group $G$, which maps every element of $G$ to the identity transformation.

1. **Problem Statement:** Prove that a finite group $G$ is simple if and only if the kernel of every non-principal irreducible character $\chi$ of $G$ is trivial, i.e., $\ker(\chi)

1. **Problem Statement:** We want to prove that a noncyclic group $G$ of order $21$ must have $14$ elements of order $3$.

1. **Problem Statement:** Prove that if $P$ is a Sylow $p$-subgroup of a finite group $G$, then $P$ is a maximal normal $p$-subgroup of its normalizer $N_G(P)$.

1. **Problem statement:** Given that $N$ and $M$ are normal subgroups of a group $G$, and $NM = \{ nm \mid n \in N, m \in M \}$ is a subgroup of $G$, prove that $NM$ is also a norm

1. The problem asks which property of a finite group ensures that every element appears exactly once in every column of its Cayley table. 2. The Cayley table of a group is a multip

1. **Problem statement:** Given that $n$ is odd, determine how many subgroups of the dihedral group $D_n$ are normal. 2. **Background:** The dihedral group $D_n$ is the group of sy

1. **Problem Statement:** Show that the finite non-solvable group $G$ satisfying the given properties is simple. 2. **Recall the definition of a simple group:** A group is simple i

1. **Problem Statement:** Explain Cauchy's theorem at an IMO (International Mathematical Olympiad) level. 2. **What is Cauchy's Theorem?** In group theory, Cauchy's theorem states

1. **Problem Statement:** Given a set $A$, let $A'$ be the set of all subsets of $A$. For example, if $A = \{1, 2, 3\}$, then $$A' = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1

1. **Problem Statement:** (i) Let $G$ be a group and $g \in G$. Define the map $T_g : G \to G$ by $T_g(x) = xgx^{-1}$ for all $x \in G$. Show that $T_g$ is an automorphism of $G$.

1. **Problem Statement:** We consider the group of isometries $\mathrm{Isom}(\mathbb{R}^n)$ under composition.

1. **Problem statement:** We are given groups $A$ and $B$, a homomorphism $\theta : A \to \mathrm{Aut}(B)$, and a set $B \times_\theta A$ with operation defined by

1. The problem is understanding how to write cube and tetrahedron rotations as permutations of vertices or faces given an axis and an angle.\n\n2. For a cube with an axis through o