A simple group is called G if G has no ordinary subgroup other than itself, and suppose f: G → H is a uniform spherical group and G is simple, prove that f is an amorphous H =< e >.

Q: Let G be an abelian group and let H= {g EG: g^n= e for some nEN}. Prove that H is a subgroup of G.

A:

Q: Let G be an abelian group with identity e. Prove that is a subgroup of G. H = {g E G|g" = e for some…

A: G is an abelian group and H is a subset of G.

Q: Let H be a normal subgroup of a finite group G and let x E G. If gcd(lxl, IG/H\) = 1, show that x E…

A: Let H be a normal subgroup of a finite group G and x∈G, Given: gcdx,GH=1. To Show: x∈H.

Q: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

A:

Q: Let ø be a homomorphism from a group G to a group G'. Prove that ker øis a subgroup of G.

A:

Q: (3) Write down all proper subgroups in the group Z3 × Z3.

A:

Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:

A: I have proved the definition of normal subgroup

Q: Theorem 3.34: Let (G, *) be a group, then (1) {e}AG. (2) GAG (3) Cent(G) AG. Proof: (H.W.)

A:

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: Let H be a normal subgroup of G and K a subgroup of G that containsH. Prove that K is normal in G if…

A: Assume that H is a normal subgroup of G, and that K is a subgroup of G containing H. Then, we need…

Q: Show that a group G = {a ∈  S5; a(1) = 1, a(4) = 4} is a subgroup of S5

A:

Q: 2. Suppose an element a E G has order 2. Prove that (a) is a normal subgroup of G if and only if a…

A:

Q: subgroup of G. Then the following statements are

A:

Q: (3) Let H be a subgroup of a group G. Let a E G. Prove that a¹Ha = {a¹ha he H} is a subgroup of G.

A:

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: If H is a finite subgroup of a group G and x ∈ G, prove that |N(H)| = |N(xHx-1)|.

A: Given: If H is a finite subgroup of a group G and x∈G. To Prove : NH=NxHx-1

Q: 1) Let G = {e,a,a',a²,a*,a,a°,a'] where a = e and let H = (a*) be a cyclic subgroup %3D %3D %3D of…

A: We have to find quotient group  G/H .

Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: Let G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

A:

Q: In the following problems, let G be an abelian group and prove that the set H described is a…

A:

Q: Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E)…

A: Given: φ:G→H is a group homomorphism and E≤H. To prove: a) φ-1(E)≤G b) If E ⊲ H then φ-1E ⊲ G

Q: 1. a) Consider the group (M(R), +). Is the subset H = {[0]a € R} a subgroup of (M(R),+)? If yes,…

A: Step 1: Step 2: Step 3: Step 4:

Q: 2. Let G = Z50 and consider it's subgroup H = (5). Find all coset representatives of 3+ H.

A:

Q: Prove that every subgroup of the quaternion group Q8 is normal. Deter- mine all the quotient groups.

A:

Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

A:

Q: What does is mean for two functions f(x) and g(x) to be orthogonal

A:

Q: 1. Consider H= (i) and K= (-1) the subgroups of Qg the quaternion group. 4) Explain why K is normal…

A:

Q: Let G and H be groups, and let 0 : G -» H be a {x E G 0(x) = e} is called the kernel of 0 and is…

A: From the given information:Let G and H be the groups

Q: If the ratio of boys to girls in tennis camp is 3:5 if there are 64 campers, how many are girls 

A: We find out how many girls as follows:  • We first need to determine the total number of parts in…

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

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

Q: 4. If p: G → G' is a homomorphism, prove that ker ý is a normal subgroup of G.

A: To prove that kernal is a normal subgroup of G

Q: f f is a homomorphism of group G into a  group G' with Kernal K, then K is a normal  subgroup of G.

A: If f is a homomorphism of group G into a group G' with Kernlal K, then we have to prove that K is a…

Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = –S, |S| = 3 and (S) =…

A:

Q: ¹) Let H be a subgroup of a group G. Define NĠ (H) : = {g €G | g Hg¯¹ = H}. Prove that N(H) is a…

A:

Q: Prove that every subgroup of the quaternion group Qs is normal. Deter mine all the quotient groups.

A: The given group is Q8=±1,±i,±j,±k. There are four non trivial subgroups namely: -1=1,-1.…