Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e, b)| be G}, where e, and ez are identity elements of G, and G, respectively, then H, and H, are normal subgroups of G, x (ii) H G, and H, G, (v) The factor (quotient) group (G, xG2)| H¡ is isomorphic to G2, and (G, x G2)| H, is isomorphic to G,.

Q: Question 7. Prove or disprove: (a) The union of two subgroups of a group (G, *) is a subgroup of G.…

A: 7)  We need to prove or disprove the given statements. a) The union of two subgroups of a group G ,…

Q: QI// Let (H, *) be a normal subgroup of the group (G, *) and we define: G/H={a*H: a € G} and we…

A: As you have posted multiple questions, so according to our guidelines we will solve only first…

Q: 3) Let G be a cyclic group then a) Z(G) = G,where Z(G)is the center of b) Every subgroup of G is a…

A: our objective is to show which options are true in the above questions.

Q: Qa\ Answer for only one. 1) Let H and K be normal subgroups of a group G, then Hn K be a normal…

A:

Q: Let f:G to H be any group homomorphism, that is, f(x)f(y) is always f(xy). It will follow that also…

A:

Q: If f is an isomorphism of a group G into a group H, prove that (f(a))n=e' if and only if an=e. Give…

A: Suppose f is an isomorphism from G to H, and let a be an element of G. Then: (f(a))^n = e' if and…

Q: Let G be a group with identity element e. Prove that for any element a in G, (a^2)^-1 = (a^-1)^2.…

A: We are given that G is a group with an identity element e. In group theory, a branch of mathematics,…

Q: (a) (Z4, +). (b) (Z5, .). (c) (Z5, +). (d) (Z4, .). (e) (Z6, .).

A:

Q: that a and xax^-1 have same order for all x belongs G.

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Q: Consider the group G = {x E R such that x 0} under the binary operation x*y=-2xy O x*x*x=4x^3…

A: Multiplication of the elements of the group elements with respect to binary operation

Q: 8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set…

A:

Q: 2. If H and K are normal subgroups of a group G and H^K = {e}, prove that G is isomorphic to a…

A:

Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: Let G be a group and H a subgroup of G. H is a normal (or distinguished) subgroup of G if and only…

A: The Center of the group is defined as, C=x∈G | ∀y∈G yx=xy. The given definition 1' is: A subgroup H…

Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

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: Prove that the group G with generators x, y, z and relations z' = z?, x² = x², y* = y? has order 1.

A: In order to solve this question we need to make the set of group G by finding x, y and z.

Q: 4. There is a theorem that says that every element g € GL(2, R) can be written, in a unique way, as…

A: Given:  g=053-12           We have to find k, a, n such that g=kan where k∈K, a∈A and n∈N.

Q: QI// Let (H, *) be a normal subgroup of the group (G, *) and we define: G/H={a*H: a E G} and we…

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Q: t the groups: G, G', G'' and the functions f:G-->G', g:G'-->G'' Prove the following statement: If…

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Q: 2) Let G be a group and H be a subgroup of G then H x = H• y -y-. xcH. true O false

A: (a) Given that G is a group and H is a subgroup of G H.x=H.yH.x.y-1=H.yy-1H.xy-1=Hxy-1∈H Hence,…

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: Suppose that f: G G such that f(x) and only if = axa. Then f is a group homomorphism if -> a = e a^3…

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Q: Question 8. (a) In the group Z10, >, find ([2]) and then find the order of the quotient group…

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Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

A:

Q: 17. Let f: be defined by f(x) = 3x - 3 Prove or disprove: fis an isomorphism of the additive group…

A: The objective is to prove or disprove :f is an isomorphism of the additive group onto itself. Group…

Q: True or False, show justification. If a, b ∈ G, where G is a group, and n ∈ Z, show that…

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Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, CG(A) = A and…

A: Given G=S3 and A=1,1 2 3, 1 3 2 The objective is to show that CGA=A, NGA=G. The definition for…

Q: 1. Consider the group U(5). (a) What is |U(5)|? (b) For each a ∈ U(5), find |a|, , and | |   2.…

A: U(n) - Set of all positive integers less than n and relatively prime to n. Then U(n) is a group of…

Q: a group and H, K be Subgroups of NG (H) = NGCH) Relate H and K? let G be G Such that %3D

A: Given: Let G be the group and H, K be the subgroups of G such that NG(H)=NG(K)

Q: E- {\,23, X= {1,2,3, 48 T= {0,213,254} find the fallowing:- 1- The goroup derivative dE), Do you (E)…

A: Given that E=1,2 and τ=∅,1,3,4,1,3,4,X Where X=1,2,3,4 And τ is a topology of X. The group…

Q: Let Dn = {1, a, ...‚ an-¹, b, ba, .. - ,ban-1} with o(a)= n, o(b) = 2, and aba = b. (a) Show that…

A: Let's first establish some properties of the group Dn:1. The group Dn is generated by a and b.2. The…