6. Show that if p is a prime number, then Z/pZ has no proper non-trivial subgroups.

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

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Q: 10. Let A be a subgroup of G, and let B be a subgroup of H. Show that A×B is a subgroup of G×

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Q: Consider the factor group G = R* /Q*. Determine which of the following elements in G are equal. 1Qx,…

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Q: Show that there is no simple group of order pqr, where p, q, and rare primes ( p, q, and r need not…

A: We know the definition of simple group, A simple group is a group whose only normal subgroups are…

Q: \(d\) is a metric on \(X\). Show that \(\rho : X^2 \to [0,\infty)\) defined by \[ \rho(x,y) =…

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Q: The group (U20, X20) = {(1, 3, 7, 9, 11, 13, 17, 19), ×20} has subgroups H = {1,3,7,9) and K =…

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Q: If pq and r are three distinct primes and G is a group of order par, then G is not simple.

A: Suppose that p,q and r are three distinct primes and G is a group of order pqr, then prove that G is…

Q: Explain why every subgroup of Zn under addition is also a subring of Zn.

A: Every subgroup of Zn under addition is also a subring of Zn as it follows the 1) Associative…

Q: In Z, find all generators of the subgroup <3>. If a has infinite order,find all generators of…

A: Since a has infinite order, the same holds for a 3 since if it would have order n < ∞, then 1 =…

Q: 2 Prove That D₂ = 8 cannot be the internal direct product of two of its proper subgroups.

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Q: 4. Consider the group A4, a group with twelve elements. Note that the positive divisors of 12 are 2,…

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Q: Show that Z has infinitely many subgroups isomorphic to Z.

A: We have to show that Z has an infinitely many subgroups isomorphic to Z.

Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: Show that in C* the subgroup generated by i is isomorphic to Z4.

A: C* is group of non-zero comples numbers with multiplication

Q: Abstract Algebra: Prove that Sylow p-subgroups for different primes can only have trivial…

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Q: Show that the elements of S9 which send the numbers 2, 5, 7 among themselves form a subgroup of S9.…

A: 1. **Closure**: If any two permutations send 2, 5, and 7 among themselves, their composition must…

Q: 8. Show that if |G| = 12 then G has a normal Sylow 2-subgroup or a normal Sylow 3-subgroup.

A: Ans-In the given question we have to show  if |g| = 12 then g has a normal sylow 2-subgroup or a…

Q: 5. List the left and right cosets of the subgroups in each of the following. Show Transcribed Text…

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Q: Determine the subgroup lattice for Zp. When p= any prime number.

A: Let H be a subgroup of group G, then H must divide GNow, ℤp=p, prime, the subgroups of ℤp are those…

Q: Prove that the intersection of two subgroups of G is itself a subgroup of G.

A: Suppose that, H1 and H2 be the sub groups of G. Also, let H=H1∩H2 . Consider one step group test:…

Q: Prove that a group of order 595 has a normal Sylow 17-subgroup.

A: