Prove that group A4 has no subgroups of order

Q: Use Cayley's theorem to find an explicit subgroup of Sg which is isomorphic to the quaternion group…

A: Given Information:Group and Quaternion group .To find:The subgroup of  isomorphic to the Quaternion…

Q: Does Sn have a subgroup isomorphic to An other than An (n ≥ 5).

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Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: Prove that A5 cannot have a normal subgroup of order 2

A: A normal subgroup is a subgroup that is invariant under conjugation by any element of the original…

Q: Prove that the subset of elements of finite order in an Abelian groupforms a subgroup. (This…

A: To Prove: The subset of elements of finite order in an abelian group forms a subgroup.

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

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Q: Find a proper subgroup of the group of integers Z under addition and prove that this subgroup is…

A: Consider 2 ℤ = {an / n∈ ℤ }This is a subgroup of ℤ .claim : 2ℤ is isomorphic to ℤ.…

Q: Prove that {(1), (1, 2)(3, 4), (1, 3)(2,4), (1, 4)(2,3)} is a subgroup of S4.

A: We use the Cayley table.

Q: No. of isomorphic subgroup of group of integers under addition is: -

A: As we know group of integers under addition is (Z,+)

Q: Show that a normal subgroup is commutative with every complex

A: Claim : A normal subgroup is commutative with every complex.

Q: a) Show that a group of order 700 has a normal subgroup of order 175.

A: To show that a group of order 700 has a normal subgroup of order 175, we will make use of the Sylow…

Q: Prove that a group of class equation 24 = 1+1+4+4+4+4+6 does not have a normal subgroup of order 4.

A: Prove that a group of class equation 24 = 1+1+4+4+4+4+6 does not have a normal subgroup of order 4.

Q: Find all subgroups of U(7)

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Q: Calculate the cyclic subgroup (15) < (Z24, +21)

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Q: Every cyclic group is a non-abelian group.

A: False, all cyclic groups are abelian group. This is true

Q: Find all distinct subgroups of the quaternion group Qs, where Q8 = {+1,±i,±j, £k} Deduce that all…

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Q: to a subgroup of Sn for n ≥ 3.

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Q: Prove that order of the subgroup divides the order of the group by the index of the subgroup…

A: See the attachment

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

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Q: Find the order of the factor group U(16)/(9)where U(16)={1,3,5,7,9,11,13,15} with operation…

A: Since , we have to find order of the factor group U(16)/&lt;9&gt; Concept: Order of factor group…

Q: Prove that S, is isomorphic to a subgroup of An+2-

A: An even permutation can be obtained as the composition of an even number and only an even number of…

Q: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order…

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Q: Construct a subgroup lattice for the group Z/48Z.

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Q: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

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Q: G = (R, +), H = {a+bv2: a,b € Z}

A: Given G = (R, +), H = {a+b√2 : a, b∈Z}. We check whether H is a subgroup of G.

Q: Determine two non trivial subgroups of U(18)

A: This question is related to Abstract Algebra

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: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

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

A: Let, H = &lt;-1&gt; Now, for any positive integer k,       -k = (-1) + (-1) + (-1) + . . . . . . . .…

Q: Show that every group of order (35)3 has a normal subgroup of order 125

A: Given, A Sylow 5-subgroup of a group of order 353 is of order 125. The divisors of 353 that are not…

Q: is a subgroup of Z_18 of order: 18 1 4 3

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Q: Prove that a normal subgroup need not to be a characteristic subgroup.

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Q: Prove C*, the group of nonzero complex numbers under multipliation, has a cyclic subgroup of order n…

A: n th root of unity is the cyclic subgroup of order n.

Q: Show that every subgroup H of the group G of index two is normal.

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Q: Draw the lattice of the subgroups Z/20Z.

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