2. Suppose an element a E G has order 2. Prove that (a) is a normal subgroup of G if and only if a is in the center of G.

Q: Defn: Let  H be a subgroup of G.  NG(H)={g∈ G | gHg-1=H } is called the normalized of H in G. a)…

A: First, we define a Normal subgroup.A subgroup H of a group G is said to be normal subgroup of G if…

Q: Let H be a subgroup of R under addition. Let K = {2ª | a E H}. Prove that K is a subgroup of R*…

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Q: (6) Define the center of a group G as the subset Z = {z EG: zg = gz for all g € G}. Prove that Z is…

A:

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: Prove that SL,(R) is a normal subgroup of GL,(R).

A: Let G=GL(n,R) be the general linear group of degree n, that is, the group of all n×n invertible…

Q: Prove that A5 cannot have a normal subgroup of order 2

A: A normal subgroup is a subgroup that is invariant under conjugation by any element of the original…

Q: (b) Prove that if N 4 H, (N is normal subgroup of H) then o'(N)<G (ø'(N) is normal subgroup of G).

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Q: (3) Let (A, +..) be a subgroup of (M₂ (Z), +,.), Then A is ideal of M₂ (Z), where A = {(a b) la, b,…

A:

Q: Let G be a group and let H and K be normal subgroups such that HnK = {e}. Let : G→G/HxG/K be the map…

A: Let  be  a  group  and let and  be  normal subgroups   such  that .Let   be  the map We have   to…

Q: 4. Recall that Z(G) = {r € G| gr = rg, Vg E G}. Show that Z(G) is a normal subgroup of G.

A:

Q: 1. Let G and G' be two groups. Let p: G G' and :G G' be two homomorphisms then prove that H = (x EGI…

A: Using the definition of subgroups, we will first show that H is a subgroup of G. And then we will…

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 G = GL2(R). {[ {[ а (a) Prove that T = | a, c, d e R, ad is a subgroup of G. d a (b) Prove that…

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Q: (4) Let N and H be subgroups of G and let N 4G. Prove that NH is normal in H.

A: Note: Since you have posted a multiple question according to guidelines I will solve first…

Q: Let G be a group and let H and K be normal subgroups such that HnK={e}. Let : G→G/HxG/K be the map…

A: The objective of this question is to prove that the kernel of the map φ: G → G/H × G/K, where φ(g) =…

Q: 2. Let G and H be any two groups. Define the set G x {e} = {(g, eμ) | gЄG} which is a subset of G ×…

A: Step 1: Step 2: Step 3: Step 4:

Q: 1. Let G₁ and G₂ be two groups, with identities e₁ and e2, respectively. Let G = G₁ x G2 and let H…

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Q: Let H be a subgroup of a group G with the property that for all a andb in G, aHbH = abH. Prove that…

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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: Let G be a group and H a subgroup of G. If [G: H] = 2 then H ⊲ G, where [G: H] represents the index…

A:

Q: Let P has order 8 is a normal subgroup of G. Show that the order of automorphism of P (Aut(P)) is…

A: We will need some facts regarding the group A_4

Q: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

Q: Let G be a group of order 2p, where p is  prime. Show that G has a normal subgroup  of order p.

A: G be a group of order 2p, where p is prime.    Claim : G has a normal subgroup of order p.

Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

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Q: K The initial tableau of a linear programming problem is given. Use the simplex method to solve the…

A: Given x1x2s1s2s3zR.H.S.14100015410100121100102-3-200010

Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: prove that Uk(n) is a subgroup of U(n)

A: The objective of this question is to prove that Uk(n) is a subgroup of U(n). Here, U(n) represents…

Q: Prove that If |G|=np with p>n and p prime, and H is a subgroup of G with order p, then: H is normal…

A: We proved by contradiction that H is normal in G  The procedure is given below