9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * None of these b-1a e H ba e H ba-1 e H

Q: 11. Let H be a subgroup of a group G and suppose that 91,92 € G. Prove that the following conditions…

A:

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: 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: 5. Let H be a subgroup of a group G. Prove that aH Ha-1 is a bijective mapping from the set of all…

A: Let H be a subgroup of a group G. Let f be a function from the set of all left cosets of H in G to…

Q: Let H be a subgroup of a group G and a, bEG. Then be aH if and only if * a-1b e H O None of these…

A: Given H is a subgroup of G. We need to find a necessary and sufficient condition for a belongs to…

Q: 24. If a and b are distinct group elements, prove that either a² + b² or a³ + b³.

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Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O b-la e H O ba e H O…

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Q: Prove or disproe: If C is a groun and a hsC then (ab)-1 h-1a-1

A:

Q: Let H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

A: We know that b∈bH                                    (1) We know that aH = bH if and only if a-1b∈H…

Q: -1 1) Let G be a group and H be a subgroup of G, let H • x,H • y, x,y € Gare two right cosets of H…

A: Any two cosets are either identical or disjoint

Q: 25. Prove or disprove that if a group G has a cyclic quotient group G/H, then G must be cyclic.

A: Consider the group G = Z2 x Z2 and H = {(0,0)}Z2 x Z2 is not a cyclic group because it cannot be…

Q: If G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G with H…

A: Given: Group G=1,2,3,4,5,6 under multiplication modulo 7. Subgroup H=1,2,4 of G. To find:   a) Right…

Q: 5. Use a direct proof to show that that in a group G, if (a + b)² = a² * b², then a * b = b * a for…

A:

Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

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Q: Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e,…

A:

Q: Is it possible to find a group operation e on a set with 0 elements? With 1 element? Explain why or…

A: The question is :: is there possible to find a group operation on a set of 0  element? Or with 1…

Q: 4*. Let a and b be elements of orders 5 and 8 in a group of order 120. List the possible orders of…

A: To list the possible orders of the subgroup generated by a and b under the given conditions

Q: If a group G has finitely many subgroups, does this mean that G is finite group? Justify your…

A:

Q: 3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b.…

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Q: If G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G with H…

A: Given G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G…

Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: Suppose (G, ◇) is a group. Which of the following is true? 1. ∀a ∈ G, a ◇ e = e ∈ G 2. ∀a ∈ G, a ◇…

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Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

A:

Q: 3. Let A and B be groups. Show that S := {(a, e) | a € A} is a normal subgroup of G:= A x B and the…

A:

Q: 1)Let G be a group and H be a subgroup of G then, for any a, b e G,a = b mod H + (a + b-1) H = H.…

A: Given: To show the correct option is true or false as follows,

Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

A:

Q: 5. Let G be a group and n e Z+ be fixed. Show that H = {a" | a € G} is a subgroup of G

A: