The class equation for a group G can be written |G| = |Z(G)| + E|CI(a), where the sum runs over representatives of the distinct non-central conjugacy classes. Recall that the conjugacy classes for S3 are {1}, {(123), (132)} and {(12), (13), (23)}. Therefore the class equation of S3 is given by | S3| = 1+ 2 + 3. a) Find the conjugacy classes for Dg and write the class equation for Dg.

Q: -) Write the Cayley table for the group U(15).

A: U(15) is the set of all elements in ℤ15={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14} ; x≡x ,such that…

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Q: Find all the non-isomorphic abelian groups of order 32.

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Q: In the group of integers Z with addition, cl(5) the conjugation class of 5 is equal to:

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Q: Give an example of a p-group of order 9.

A: Given, Give an example of a p-group of order 9.

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Q: Determine the conjugacy classes of Ag and the class equation for Ag.

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Q: 9. Classify the following groups in the sense of the Fundamental Theorem of Finitely Gener- ated…

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A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

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A: Ans: (d) not true.For details please see the explanation.If you have any problem then let me know in…

Q: 5. Let G be a finite group, and let class (₁), ..., class(xn) be all of the distinct non- singleton…

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Q: Find the order of the given factor group (Z3 x Z6)/

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Q: Prove that if (ab)' = a*b² in a group G, then ab = ba.

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Q: a) Check whether G = (3n, ·) is a group, where n ∈ Z. b) Let p1 (x) = x3 + 1 and p2 (x) = x4 + x.…

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Q: The order of identity element equal itself in any group

A: .

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Q: Explain why each of the following could not be the class equation of a group G. i) |G| = 3 + 3 + 3…

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A: The concepts have been explained with examples.

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