5. For each a in the group G, define a mapping ta: G→ G by ta(x) = axat. Prove that to is an automorphism of G.

Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

Q: Prove that any group with three elements must be isomorphic to Z3.

A: Let (G,*)={e,a,b}, be any three element group ,where e is identity.  Therefore we must have…

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: Let G be a group. Using only the definition of a group, prove that for each a E G, its inverse is…

A:

Q: (a) Let p: G → H be a group homomorphism. Show |p(x)| < |x| for all x E G.

A:

Q: Suppose o is a homomorphism of groups fom G onto H. If H has an element of order 100, then show that…

A:

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

A: We have given a map ,  ϕ : A → B , where A = ℝ , +   ,  B = ℝ* , · such that  , ϕx = 2x We know that…

Q: What is the automorphism group of ?A ut (Z3, +)

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

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

A:

Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

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: integer 16. If EG→His a surjective homomorphism of groups and G is abelian, prove that H is abelian.…

A:

Q: is] Let G and H be groups, and let T:G→H_be Isomorphism. Show that if G is abelian then H is also…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

A: Given 'x' be an element of a group G and |x|=n. As G be a group , inverse of each element of G must…

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

A:

Q: For the group homomorphism : Zs → Z, defined by ø([r]) = [æ]² for all [r] € Z, find the kernel and…

A:

Q: Let G and G' be groups. (a) If p: G → G' is a function, what is required for p to be a homomorphism?…

A:

Q: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

A:

Q: Let G be a group and define the map ø : G → G by $(9) = g¬1. Show that o is an automorphism if and…

A: The solution is given as

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: 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: Suppose that o is a homomorphism from a finite group G onto G and that G has an element of order 8.…

A:

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: (a) Prove that every element of Q/Z has finite order. (b) Given two groups (G, .) and (H, *).…

A: This question is related to group theory.

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: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

A:

Q: Show that any automorphism of GF( pn) acts as the identity onGF( p).

A:

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: Find a homomorphism from the group (Q, +) to (R, +). Find a one-one map if exist from Q to R that is…

A: Given :  (Q, +)  and (R, +) be the given additive groups. To find :  A homomorphism from (Q, +) to…

Q: Prove that An even permutation is group w.y.t compostin Compostin function.

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: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

A:

Q: Determine whether an onto homomorphism between the groups D6 and D3 + Z2 exists.

A:

Q: Let x be in a group G. If x' - e and x* - e , prove that x - e and x' = e

A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…

Q: Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X…

A: Solution: Let us consider (G, .) be a group and B be a subgroup of the group G. Now for any g∈G, the…

Q: 2* Let f G H be a group homomorphism. Prove that if x E G and n is a natural number then f(x)= f(x)"

A: To prove the required property of group homomorphisms

Q: 1. If H be a subgroup of group G, then the relation on G defined by a ~ b if and only if abe H, for…

A:

Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

A:

Q: 1. Consider the groups (R+, ) and (R,+). Then R* and R are isomorphic under the mapping $(x) = log10…

A: We use the definition of cosets, isomorphisms to answer these questions. The detailed answer well…

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}

Q: If a group G is isomorphic to H, prove that Aut(G) is isomorphic toAut(H)

A: We have to prove, If a group is isomorphic to H, then Aut(G) is isomorphic to Aut(H).

Q: 2. Let H and K be subgroups of the group G. (a) For x, y E G, define x ~ y if x = hyk for some h e H…

A: