The image contains several mathematical problems focused on group theory and geometry. I’ll summarize and interpret the problems from the image for detailed exploration:

Problems:

1. Geometric Problem (Score = 15):

   • Given two points $A(1,2)$ and $P(a,b)$, determine $P$ such that $\sigma_A(P) = (3,6)$.

2. Inverse Elements in a Group (Score = 20):

   • Prove that the inverse of the composition of two reflections is the composition of their individual inverses.

3. Transformation Groups (Score = 30):

   • Prove that $(H, \circ)$, the set of transformations $\sigma_A$, is a group under composition.

4. Subgroup of a Symmetry Group (Score = 35):

   • Given $G$, the set of all symmetries of a vector space $V$, and $H \subset G$ as a subset containing specific transformations, prove that $H$ is a subgroup of $G$.

Would you like me to provide detailed solutions to one or all of these problems? Let me know if you'd prefer step-by-step explanations or if you'd like me to focus on a specific one!

Related Questions:

1. What are the fundamental properties required to form a group?
2. How is the composition of transformations like reflections defined mathematically?
3. What are the necessary conditions for a subset $H$ to qualify as a subgroup of a group $G$?
4. Can symmetry groups of geometric objects form infinite groups? If yes, under what circumstances?
5. How does the operation of reflection work in two-dimensional geometry?

Tip: For proving $H$ is a subgroup of $G$, use the subgroup criteria: closure, identity, and inverses within $H$.