Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = –S, |S| = 3 and (S) = Zn. %3D (a) Prove that n is even number. (b) If n > 6, prove that Cay(G, S) is a normal Cayley graph.

Q: Let p be an odd prime number. Let Dp be the diehedral group of order 2p. Find |Aut (Dp)|.

A: Dp is the Diehedral group of order 2p, where p is an odd prime number.  We have to find AutDp.

Q: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

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Q: IL Show thai al subgroup of erder 125. 18. Show that there are no simple groups of order 255 =…

A: By using syllow theorems the solution is given as follows :

Q: Theorem 3.34: Let (G, *) be a group, then (1) {e}AG. (2) GAG (3) Cent(G) AG. Proof: (H.W.)

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Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: center of G is defined to be Z(G) = {a E G | ag = ga for all ge G}. (a) Show that if X C G is a…

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Q: Consider the group A5. (a) Compute a = (1,2, 3)(3, 4, 5) and ß = (1, 2, 3)(2, 3, 4). (b) List…

A: Solution given below

Q: 8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set…

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Q: (14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd…

A: The order of every subgroup of a finite group G is divisor of the order of G.

Q: 6.9. Let G = Cn be a cyclic group of order n, and let X = {x₁,...,n} be a set containing n elements.…

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Q: describe all the elements in the cycle subgroup GL(2,R) generated by the given 2 x 2 matrix [0 -1]…

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Q: 16* Find an explicit epimorphism from S5 onto a group of order 2

A: To construct an explicit homomorphism from S5 (the symmetric group on 5 symbols) which is onto the…

Q: (5) Let (G, *) be a group with a² = e for every a € G. Show that G, a*b = b* a (that is, show that…

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Q: Let G = D10 × Z/7Z, where, as usual, D10 = (a, b | a³ = b² = e, ba = a*b). Let H ((b, 1)) and K =…

A: Since you have posted a question with multiple subparts, we will solve the first three subparts…

Q: = n and o(b) = m Let G be a group and suppose a, b E G satisfy ab = ba. Assume that o(a) : with (n,…

A: Given G is a group, such that a, b∈G satisfy ab=ba. Also, we have oa=n and ob=m with n, m=1.   To…

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: (b) Find all bEG with order /b| = 7 bEG. are: (c) Construct in Ga subgroup H with index.(H) = 7

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Q: Show that the group (U7, *) is isomorphic to {Z6, +}. how that there is an isomorphism between the…

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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: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

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Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

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Q: Employ the following methods to find the maximum of f(x) = 2x – 1.75x² +1.1x – 0.25x* Newton's…

A: The function given is as follows: fx=-0.25x4+1.1x3-1.75x2+2x..............................(1) The…

Q: Let G be a group and H ≤ G. (a) Prove for any a € G, aHa-¹ ≤G. (b) Let x E G and Prove C (x) ≤ G.…

A: Vital instruction : According to Bartleby Guideline , i can solve at most first three subparts if…

Q: 9. T: R2 R2 first performs a horizontal shear that trans- →>> forms e₂ into e2 - 2e₁ (leaving e,…

A: