If a and b are arbitrary distinct elements of a  group G and H is any subgroup of G, then           Ha=Hb⇔ab\power{-1}∈H           aH=bH⇔b\power{-1}a∈H.

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: Let G be a group. Suppose G=Z6xZ4. State all subgroups of G. Categorized(classify) the…

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Q: Let a and b be elements of a group G and H and K be subgroups ofG. If aH = bK, prove that H = K.

A: Given: a and b are the elements of  group G and H respectively and K is a subgroup of G. To Do: If…

Q: Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group…

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Q: Let G = <a, b | a6 = b3 = e, b-1ab = a3>. How many elements doesG have? To what familiar group…

A: Given set is G = <a, b | a^6 = b^3 = e, b^-1ab = a^3>.

Q: Question 2. Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G…

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Q: Prove that if (ab)' = a*b² in a group G, then ab = ba.

A: Given,ab2=a2b2To prove: ab=ba

Q: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

Q: G

A: According to the question, Let's start by defining the dihedral groups D3 and D5,  So, D3 is the…

Q: Decide if each of the following sets H is a subgroup or not of the group G = R XR, where the group…

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Q: Find the regular permutation group isomorphic to the multiplicative group G = {1, −1, i, −i}.

A: We will find the regular permutation group isomorphic to the multiplicative group…

Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

A: It is given that, H be a subgroup of S5 which contains a 5-cycle and a 2-cycle. We have to show…

Q: 3. Let G be a group, HAG, KAG,HnK = {e}. Show that G= G/H x G/K.

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Q: disprove: If G is a group and H, K < G are two subgroups, then (HK)≤ G.

A: