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Q: 1. Let p: G → H be a group homomorphism. Show: (c) If K is a subgroup of G, then ) p(K) is a…

A: (c) To check a subset ϕ(K) ⊆ H is a subgroup of H, It is need to verify three things: (i) Let K be a…

Q: 9. Let G be a cyclic group with |G| = 24 and let a be an element of G. If a - cidentity) and a" -…

A: Let G be a group of order 24 and let a∈G. Given a8≠e and a12≠e. Prove G=&lt;a&gt;.

Q: Question 2. The subgroup generated by S could have been defined a second way, as the set of all…

A: The subgroup generated by a set in a group , where represents the subgroup generated by and is…

Q: 22. Let 0 G → H be a group homomorphism, and let K = ker = {a € G|0(a) = EH}. Show that gK = Kg for…

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Q: True or False: If H is a subgroup of G and G/H is cyclic, then G is cyclic. (a) True (b) False (c)

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Q: empity subsel Hef a Ha,bE H. G. Pawe that the intersection Hhk is also a subaroup. Eseercise I (a)…

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Q: Let G = : a – b = c – d, a,b, c, d E R Show that G is a group under (the usual) matrix addition.

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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…

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Q: True or False: (a) 3Z = 9Z (b)Let p be a prime number. Then Zp × Zp = Z,² (c) Every subgroup of a…

A: We have to state whether the following statements are true or false. We have  and . We have to show…

Q: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

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Q: 3) Let G be a group. Show that if Aut (G) is cyclic, then G is abelian.

A: Solution is given below

Q: Question 4. Suppose that H and K are subgroups of a group G and there are elements a, b e G such…

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Q: Let sulgpoups Haud K have finite indices k aud e, their in a group G. Peove that the index of…

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Q: Cyclical group question: Let G be a group. We say that G is cyclic if G there is one element ge G…

A: G is a cyclic group generated by an element g, i.e., G=gn n∈ℤ.

Q: Let G=U(18) and H={1,17} be subgroup of G. The number of distinct left cosets a of H in G is: * 3.

A: Given G=U(18)  H ={1,17} We need to find the number of distinct left cosets of H in G

Q: Aut (2) E Z..

A: