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A: As per our guidelines we are suppose to answer only one ques. Answer of question 1 is as follows:
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A: Given question: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?
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- Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.True or False Label each of the following statements as either true or false. 6. Any two groups of the same finite order are isomorphic.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
- 9. Find all homomorphic images of the octic group.True or False Label each of the following statements as either true or false. In a Cayley table for a group, each element appears exactly once in each row.Label each of the following statements as either true or false. The Generalized Associative Law applies to any group, no matter what the group operation is.
- True or False Label each of the following statements as either true or false. The order of an element of a finite group divides the order of the group.Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.Let H and K be arbitrary groups and let HK denotes the Cartesian product of H and K: HK=(h,k)hHandkK Equality in HK is defined by (h,k)=(h,k) if and only if h=h and k=k. Multiplication in HK is defined by (h1,k1)(h2,k2)=(h1h2,k1k2). Prove that HK is a group. This group is called the external direct product of H and K. Suppose that e1 and e2 are the identity elements of H and K, respectively. Show that H=(h,e2)hH is a normal subgroup of HK that is isomorphic to H and, similarly, that K=(e1,k)kK is a normal subgroup isomorphic to K. Prove that HK/H is isomorphic to K and that HK/K is isomorphic to H.