8. Let G be a simple group of order 60. Then (a) Ghas six Sylow-5 subgroups (Ъ) Ghas four Sylow-3 subgroups (c) Ghas a cyclic subgroup of order 6 (d) G has a unique element of order 2

Q: Find the (right) cosets for the following subgroups of the stated groups: (i) The subgroup H =…

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Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

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Q: Suppose G- (a) is a cyclic group of order 16. Let H (a* ) be the cyclic subgroup generated by a (a)…

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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: 19. Prove that the group (Z/11Z)* is cyclic, and find a generator.

A: We have to prove that the group (Z/11Z)* is cyclic, and find a generator.

Q: 4.6. (a) Let G be a cyclic group of order 6. How many of its elements generate G? Answer the same…

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Q: 4. Let G be a group of order 14. Using Lagrange's theorem, show that every subgroup of G either…

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Q: List all of the subgroups of the group (Z20. +). empty subset of G tha

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Q: Let G = A4, the alternating group of degree 4. (a) How many elements of order 3 does G have? (b)…

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Q: 7. (a) What is the order of (1, (123)) in Z4 × S3 ? (b) Write down all the distinct cosets of the…

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Q: 18. Let G be the group (Z/7Z)x. Determine all the cyclic subgroups of G.

A: In this context,  represents the multiplicative group of integers modulo 7 that are relatively…

Q: b. Show that a cyclic group of order 6 has a cyclic group of order 2 as a homomorphic image.

A: To show that a cyclic group of order 6 has a cyclic group of order 2 as a homomorphic image, we need…

Q: the group (a) find all possible orders of elements in G; (b) for each possible order of an element…

A: (ℤ/364ℤ,+) is isomorphic to (ℤ364,+). All possible orders of (ℤ364,+) are all possible divisors of…

Q: Draw a diagram too please!

A: Step 1: Step 2: Step 3: Step 4:

Q: 2. What is the cyclic subgroup of Z generated by -1 under +?

A: Let, H = <-1> Now, for any positive integer k,       -k = (-1) + (-1) + (-1) + . . . . . . . .…

Q: 6. Let G be GL(2, R), the general linear group of order 2 over R under multiplication. List the…

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Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 23 4) and b…

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Q: 5. Let G be a group. Let H₁ CH₂ C H 3 C · of subgroups of G, and let H∞ = Un=1 Hn. be an infinitely…

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Q: 10 2) (i) Prove that H = (0, 2, 4, 6, 8) is a subgroup of Z (ii) Prove that H is a normal subgroup…

A: Answer

Q: Let G be a group of order 40 whose subgroups include H and K, of orders 4 and 5, respectively. to…

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Q: Let G = (a) be a cyclic group of order 80. (a) How many subgroups of order 10 are there in G? List…

A: concept:1 given G=<a> be a cyclic group of order 80. We know that  Any cyclic group of order n…