Let f:GH be a group homomorphism such that |G|= 10 and |H|= 12. Which one of the following statements is true: Select one: O af isone-to-one. O b.f is onto. O c. Ker(f)=2 O d. None of the statements above are true.

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

Q: Suppose thatf:G G such that f(x) and only if - axa. Then f is a group homomorphism if O a^2 = e a =…

A: See solution below

Q: 3. Define an operation on G = R\{0} x R as follows: (a, b) (c,d) = (ac, bc + d) for all (a, b),…

A: 3. Define an operation * on G=ℝ\{0} ×ℝ as follows: (a,b)*(c,d)=(ac, bc+d) for all (a,b), (c,d) ∈G…

Q: If a is a group element, prove that every element in cl(a) has thesame order as a.

A:

Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: Given information n=10.

Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: For positive integer n, let Cn denote a cyclic group of order n. If G is an abelian group of order…

Q: Prove that the trivial G is faithful if and only if G = {1G} representation of a nite group

A:

Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

Q: 3. Let m and n be positive integers such that m divides n. (a) Prove that the map y: Z/nZ Z/mZ…

A: Given m and n are two positive integers such that m divides n. To prove the map φ:ℤ/nℤ→ℤ/mℤ such…

Q: Consider the map o : G¡ → G2 defined by: 9(a) = a-! %3D (a) Does o define group homomorphism? (b)…

A: Property of homomorphism of groups....

Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

A:

Q: Suppose thatf:G G such that f(x) = axa. Then f is a group homomorphism if and only if O a^2 = e a^4…

A:

Q: Let G be a group. Define ø : G → G by ø(x) = x-1 for all x E G. (a) Prove that ø is one-to-one and…

A:

Q: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

A:

Q: Let (G, *), (G', *' and (G", ") be groups, and let :G G' and : G' → G" be isomorphisms. Prove that t…

A: We will use the definition of isomorphism to prove that the composition of two isomorphisms is again…

Q: Consider the group G={x € R such that x#0} under the binary operation *: Th identity element of G is…

A: Solution: Since for any x,y∈G, the operation * is defined as x*y=-2xy The identity element is e=-12…

Q: If f: G to H is a surjective homomorphism of groups and G is abelian, prove that H is abelian.

A: As we know that a group homomorphism f:G to H is a map from G to H satisfying:

Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: Need to evaluate all the abelian group of order 36.

Q: a) Prove that the mapping from U(16) to itself given by x→x Is an Automorphism b) Find the group SG,…

A:

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A:

Q: Prove that each of the following sets, with the indicated operation, is an abelian group. (a) (R, *)…

A:

Q: For each element a in the group G, define the mapping ka: G → G by ka(x) = xa-'for all x in G. %3D…

A: Since, you have posted multiple questions, but according to guidelines we will solve first question…

Q: Suppose that f: G → G such that f(x) = axa. Then f is a group homomorphism if and only if a = e O…

A: From the condition of group homomorphism we can solve this.

Q: Determine whether the following statements are true or false. 1. There exists an injective…

A: 1. There exists an injective homomorphism from D10 to ℤ10. Suppose there exists an injective…

Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…

A: Given that G,·and H,* are two groups.

Q: Given that A and B is a group. Find out if þ: A→B is a homomorphism. If it is a homomorphism, also…

A: Group homomorphism is nothing but a function defined between two groups. The function must be closed…

Q: | Suppose that p: U15 → U15 is an automorphism. Define H = {x € U15 |¢(x) = x¬1}. Which of the…

A:

Q: Suppose that f: G → G such that f(x) = axa. Then f is a group homomorphism if and only if a = O a^4…

A:

Q: 2. Suppose M and D are isomorphic groups. Prove that Aut(M) is isomorphic to Aut(D).

A:

Q: Let f.g:(G,.) (G')and h(m) f(m). g(m), Vm € G, them h is group homomorphism if f is homomorphism O f…

A:

Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

A:

Q: Suppose that f:G-G such that f(x)- axa. Then fis a group homomorphism if and only if O a*2e O an4e O…

A: Here we will evaluate the required condition.

Q: Let f: Z→Z be a group homomorphism such that f is one-to-one, then which of the following statements…

A: hence option (c) is the correct option

Q: Let G = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y…

A:

Q: Suppose that f: G → G such that f(x) = axa?. Then f is a group homomorphism if and only if O a^4 = e…

A: Given that f from G to G is a function defined by f(x)=axa2 Then we need to find a necessary and…

Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

A:

Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

A:

Q: For each of the following values of n, describe all the abelian groups of order n, up to…

A: We use the following result here. Result: Let n=pq for some prime numbers p and q with p&lt;q. If p…

Q: Let G = {2'3m5n : l, m, n E Z}. G is a group of rational numbers under the usual multiplication.…

A:

Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: 2. For an arbitrary set A, the power set P(A) = {X | X C A}, and addition in P(A) defined by X+ Y =…

A: Recall the following. Definition of group: Suppose the binary operation * is defined for element of…

Q: Let S = {x €R | x + 3}. Define * on S by a * b = 12 - 3a - 3b + ab Prove that (S, *) is a group.

A: The set G with binary operation * is said to form a group if it satisfies the following properties.…

Q: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

A: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

Q: 7 Suppose that a E G. For each y in the conjugacy class of a, let G(a y)= {g E G: gag= y} be the set…

A: To prove that the given subset G(a -&gt; y) is a coset of the centralizer of a in G.

Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

Q: Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism : G' → G such that…

A:

Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.

Q: 1. AUT(G) := {p: G → G|p is an automorphism} Prove: (AUT(G),º) is a group (under composition)

A: Given: Aut(G)={ φ:G→G| φ is an automorphism}