One to one and onto mapping

Let G be the real numbers under addition G' the positive real numbers under multiplication. Show that the mapping... is one to one and onto. Please see attached for the full question.

Abelian groups

Let G ---> H be a group homomorphism and... Show that if... then Φ[G] is albelian. Please see attached for full question.

Identity element in a group

Let Φ be a homomorphism of group G into a group G'. Show that if e is the identity element of G, then Φ(e) is the identity element e' in G'.

Associative & Commutative Rule

Prove that addition modulo n, written +n is: 1) associative 2)comutative there are two ways to prove these properties. each way requires a definition or two: 1) for n≥2, 0≤a, b≤n+1 a+n(written as a power in a corner downside, but dont know how to put it tho) b={condition 1 - a+b if a+b ...continues

1. This question is concerned with subgroups of the group S5 of symmetries (or permutations) on the set {1,2,3,4,5}, a group with 120 elements. (a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6 and give examples of each of these. Explain why this group does not have cyclic subgroups of any order. (b) By considering the symmetry groups of appropriate geometric figures, give examples of : (iv) a subgroup of order 4 that is not cyclic; (v) a subgroup of order 6 that is not cyclic; (vi) a subgroup of order 8. (c) By considering those permutations that fix one element, or, otherwise, give an example of a subgroup of order 24 and another of order 12. [You need not list all the elements of these groups, but you should explain clearly which elements constitute each subgroup.] (d) List the potential orders of subgroups of S5 (other than S5 itself), according to Lagrange’s theorem, in addition to those already considered in this question. Give an example of a subgroup of one of these orders. 2. (a) Which of the following sets are groups under the specified binary operation? In each case, justify your answer. (ix) Z, the set of integers, under operations * defined by a*b = a + 2b (x) R*, the set of non-zero real numbers, under the operation × defined by x×y = 5xy (xi) The set {3,6,9,12} under multiplication modulo 15. (xii) The set of matrices {(1, p;0,1)/pЄZ} under matrix multiplication. (b) G is a group of real functions with domain and co-domain the non-negative real numbers, i.e. functions [0,∞) → [0,∞). The group operation in G is function composition. If one of the elements of G is the squaring function, f, defined by f(x) = x2 , explain why G must be an infinite group. 4. (a) Define the motion of conjugacy as it applies in a general group. Prove that the inverses of a pair of conjugate elements are also conjugate. Prove that conjugate elements have the same order. The remainder of this question concerns the group G, whose Cayley table is as follows: e a b c d f g h i j k l ----|-------------------------------------------------------------------------------------------------- e | e a b c d f g h i j k l a | a b e d f c h i g k l j b | b e a f c d i g h l j k c | c f d e b a j l k g i h d | d c f a e b k j l h g i f | f d c b a e l k j i h g g | g h i j k l e a b c d f h | h i g k l j a b e d f c i | i g h l j k b e a f c d j | j l k g i h c f d e b a k | k j l h g i d c f a e b l | l k j i h g f d c b a e (b) Determine the inverse and the order of each of the elements of G. (c) Simplify each of the following: (i) acb (ii) bca (iii) ajb (iv) bja (v) gcg (d) Given that the only element conjugate to g is g itself ( you need not prove this), determine the conjugacy classes of G. (e) Find H, a normal subgroup of G having three elements. Identify the elements of the quotient group G/H and determine its isomorphism type.

Modern Algebra Group theory Symmetric Groups Permu ...continues

Group Theory : Homomorphisms, Kernels, Isomorphisms and Fields

(a) If G1 and G2 are groups, define what it means to describe a function h:G1 -> G2 as a homomorphism. (b) If h: G1 —> G2 is a homomorphism, define the kernel of h.Prove that the range of h is a subgroup of G2 , and that the kernel of h is a normal subgroup of G1. (c) Let G be the group of 2x2 real matrices under additi ...continues

Group Theory : Abelian Groups and Subgroups

3 (a) (i) Let G=Z12(sub12 don’t know how to put it), the group of integers modulo 12. Prove that H= {0, 6} AND K= {0, 4, 8} are subgroups of G. Calculate the subset H+K formed by adding together all possible pairs of elements from H and K, i.e. H+K= {h+kh is a subgroup of H, k is a subgroup of K} Prove that this is also a sub ...continues

Group Theory : Symmetry Groups, Cyclic Subgroups and Permutations

Thisquestion is concerned with subgroups ofthe group S5 of permutations on the set {1,2,3,4,5} , a group with 120 elements. (a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6, and give examples of each of these. Explain why this group does not have cyclic subgroups of any other order. (8 marks (b) By co ...continues

Group Theory : Conjugacy, Cayley Table, Subgroups and Quotient Groups

Define the notion of conjugacy as it applies in a general group. Prove that the inverses of a pair of conjugate elements are also conjugate. Prove that conjugate elements have the same order. (6 marks) The remainder of this question concerns the group G , whose Cayley table is as [TABLE] (b) Determine the inverse and th ...continues

Semi-Direct Products, S4 Groups and Homomorphisms

Let G = (Z/3Z)^4 SemiDirectProduct S_4 be the semi-direct product of (Z/3Z)^4 and S_4. Here S_4 acts on (Z/3Z)^4 by permutating the coordinates. Hint: Given H1, H2 an element in (Z/3Z)^4 and K1, K2 an element in S4. The semi-direct product is given by the operation (H1, K1) * (H2, K2) = (H1 + K1(H2), K1 * K2) A) Find the C ...continues