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 a simple group cannot have a subgroup of index 4.29. Prove that there is no simple group…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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: 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: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: Prove that in F7 the cyclic subgroup generated by x is a normalsubgroup.

A: Given: F7We need to prove that the cyclic subgroup generated by x in F7 is a normal subgroup.

Q: Explain why a group of order 4m where m is odd must have a subgroupisomorphic to Z4 or Z2 ⨁ Z2 but…

A: Given: Group of order 4m where m is odd.We need to show that :a) Given group must have a subgroup…

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: Write the Caley table of each of the subgroups of each abelian group of order 8.

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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.

A:

Q: prove that Z(S_3) is the trivial subgroup, that is, the center of the permutation group S_3 is the…

A: prove that Z(S_3) is the trivial subgroup, that is, the center of the permutation group S_3 is the…

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: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let H be a subgroup of S5. we see that, HA5=HA5H∩A5 It follows that, A5≤HA5H∩A5≤S5 Since, S5=2A5, so…

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

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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 the lattice of subgroups of D₁.

A: This question from group theory , related to topic lattice of subgroup.

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: Length is 3x − 4, area is 6x^4 − 8x^3 + 9x^2 − 9x – 4

A: On solving this we will get

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

A: