Q. How many indistinguishable necklaces can be made from beads of 4 colors with exactly 9 beads of each color where each necklace is of length 16?

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Q. Let H be a finite group. The order of Sylow p-subgroup of H for every prime factor p with multiplicity 9 is?

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Q. If the 4 sides of a square are to be colored by colors. How many different colourings with 50 colours are there if two arrangements that can be obtained from each other by rotation are identical?

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Q. Suppose that H be an X-set and suppose that a∼b and |Xₐ|=|Xᵦ|, the which of the following is true?

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Q. _______ characterizes the properties of distributive lattices.

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Q. If we take a collection of {∅, {2}, {3}, {5}} ordered by inclusion. Which of the following is true?

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Q. Which of the following is not an abelian group?

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Q. Suppose (2, 5, 8, 4) and (3, 6) are the two permutation groups that form cycles. What type of permutation is this?

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Q. Let M be an 4×4 matrix with real entries such that Mᵏ=0, for some k≥1. Find the determinant value of (I+M), where, I be the 4 x 4 identity matrix.

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Q. If 54th row of a 67th row matrix is linearly independent with each other then find the rank of the matrix.

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Q. If Y⁹⁸ (a raised to the power of 5) = 0 and Y is a 97-square matrix. Determine the value of Y⁹⁷.

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Q. Suppose, M is a lower triangular matrix with all diagonal entries zero. The resultant matrix of M+I will be ___________

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Q. Let X be a n-square matrix such that Y = X + 8I. Which of the following property will exist?

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Q. Let (z, *) is a group with x*y=x+y-2 then inverse of x is ___________

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Q. The dihedral group having order 6 can have degree _____________

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Q. Suppose Kₘ={P∈Sₘ|, |P| is odd prime}. Determine the set for which m≥3 Kₘ a subgroup of Sₘ.

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Q. Consider an integer 23 such that 23 >= 3p for a 2p-cycle in a permutation group, then p is ___________

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Q. A group G of order 20 is __________

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Q. An element of a commutative ring R(1≠0) is nilpotent if __________

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Q. All the rings of order p2 is ____________

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