Prove that the 2nd smallest non-abelian simple group is of order 168.

Q: group of order 126 cannot be simple

A: a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: Write the Caley table of all abelian groups of order 8

A: We have to write cayley table of abelian group of order 8.

Q: Prove that an Abelian group with two elements of order 2 must have a subgroup of order 4.

A: Abelian groups are groups where elements are commutative. That is, ab=ba

Q: Find all the non-isomorphic abelian groups of order 32.

A:

Q: give an example of a finite, non-cyclic abelian group containing a container of order 5

A: Take the abelian group G=Z5×Z5 of order 25 whose every element (except identity) is of order 5 and…

Q: Give an example of a non-Abelian group that has exactly threeelementsof finite order.

A: Let G=Z3⊕Z be an additive group where e=0,0. Let x∈G then it can be observed that x=a,b where a∈Z3,…

Q: Prove that  a group of order 1000 cannot be simple

A: Sylow's theorem: Let n be a positive integer that is not prime, and let p be a prime divisor of n.…

Q: Find the non-isomorphic abelian groups of  order 360.

A: We have to find the non -isomorphic abelian group of order 360.

Q: Use the theory of finite abelian groups to find all abelian groups of order 4. Explain why there are…

A: we know that if G is finite group whose order is power of a prime p thenZ(G) has more than one…

Q: (1) State the structure theorem for finite generated Abelian group. Using this theorem to classify…

A: To find the all finite generated Abelian group of order 120: Structure theorem for finite generated…

Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

Q: Find all abelian groups (upto isomorphism)  of order 67

A: We have to find all abelian groups (upto isomorphism) of order 675.

Q: Define a binary operation on R\ {0}by x*y = . 2 Prove that this set with this binary operation is an…

A:

Q: Find all abelian groups upto isomorphism  of order 4 and 6.

A: We have to find the all abelian groups upto isomorphism of order 4 and 6.

Q: Prove that a group of order 45 is abelian.

A:

Q: Determine the class equation for non-Abelian groups of orders 39and 55.

A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.

Q: List six examples of non-Abelian groups of order 24.

A: The Oder is 24

Q: Give an example of two non-homomorphic finite abelian groups that have the same number of elements…

A: We need to find an example of two non holomorphic finite abelian groups which have the same number…

Q: Prove that a group of order pq is abelian where p = 9 and q= 5.

A: We have to prove that a group of order pq is abelian where p = 9 and q= 5.

Q: Prove that a group of order 45 is abelian.

A: We have to proof that a group of order 45 is abelian.

Q: Up to isomorphism, list all abelian groups of order 1350.

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Q: not

A:

Q: Write the Caley table of all non-abelian groups of order 8.

A:

Q: Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic…

A: