State the formal definition of a group a subgroup and a grou

State the formal definition of a group, a subgroup, and a group homomorphism. Give three examples of each definition. Show that the subset G of C*: = C \\ {0} defined by G:= {e^2pi i | q Q} is a group. Can you think of any non-trivial subgroups of G. Let G be the same group as in problem 2. Let S C* be the set of all solutions (real and complex) to the quartic equation x^4 = 1 (i.e., S should have exactly four distinct elements). Find a surjective group homomorphism f from G to S and describe the kernel, ker(f), of your group homomorphism. Let A be an arbitrary matrix in GL_n(R). Let f_A: SL_n(R) rightarrow SL_n(R) denote the function defined by f_A(B):= AB A^-1. Show that f_A is a well-defined group homomorphism. Describe the groups ker(f_A) and (f_A).

Solution

1. Group: A group is a set, together with an operation that combines any two elements a and b to form another element, denoted a • b or ab.

eg.: set of integers which consist ..., 4, 3, 2, 1, 0, 1, 2, 3, 4, ...

Subgroups: A subgroup is a group H contained within a bigger one, G.[30] Concretely, the identity element of G is contained in H, and whenever h1 and h2 are in H, then so are h1 • h2 and h11, so the elements of H, equipped with the group operation on G restricted to H, indeed form a group.

eg:G={1,2,3,4,5,6,7,8,9,0} then X={1}, Y={2,3,4,5,6,7} are subgroups

Group homomorphisms
Group homomorphisms are functions that preserve group structure. A function a: G H between two groups (G, •) and (H, ) is called a homomorphism if the equation:

a(g • k) = a(g) a(k) holds for all elements g, k in G