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: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

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Q: Explain Special Linear Group is a Normal Subgroup of General Linear Group?

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Q: Show that SL(n, R) is a normal subgroup of GL(n, R). Further, by apply- ing Fundamental Theorem of…

A: Suppose, ϕ:GLn,R→R\0 such that ϕA=A for all A∈GLn,R Now, sinceA∈GLn,R if and only if A≠0 Now, we see…

Q: Prove that a group of order 375 has a subgroup of order 15.

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Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

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Q: Every

A: We will be using sylow's theorems and it's consequences to arrive at the conclusion that statement…

Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: 9. Prove that a group of order 3 must be cyclic.

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

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Q: Prove that an Abelian group of order 2n (n >= 1) must have an oddnumber of elements of order 2.

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Q: Show that the center of a group of order 60 cannot have order 4.

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Q: Give an example, with justification, of an abelian group of rank 7 and with torsion group being…

A: consider the equation

Q: Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required…

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Q: 9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no…

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Q: Every group of order 4 is cyclic. True or False then why

A: Solution

Q: The intersection of two normal subgroups is always normal sub group. (True/False)

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Q: 17. Show that every group of order (35)° has a normal subgroup of order 125.

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Q: Describe all the elements in the cyclic subgroup of GL(2,R) generated by the given 2 × 2 matrix. -1

A: Let A=0-1-10∈GL2, R Then the cyclic subgroup generated by A denoted by A=An| n∈Z

Q: Prove that every subgroup of the quaternion group Q8 is normal. Deter- mine all the quotient groups.

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Q: find Aut(Z30).  Use the FUndamental Theorem of Abelian Groups to express this group as an external…

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Q: 1. Prove that a subgroup which is generated by W-marginal subgroups is itself W-marginal.

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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 9…

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Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

Q: (8) If H1, H2 are 2 subgroups of G, prove that H1 N H2 is also a subgroup of G. If further assume…

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Q: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

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: 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: Show that any group of order less than 60 is cyclic

A: This result is not correct. There is a group of order less than 60 which is not cyclic.

Q: Prove that

A: To prove: Every non-trivial subgroup of a cyclic group has finite index.

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

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Q: Determine the class equation for non-Abelian groups of orders 39and 55.

A: We have to determine the class equation for non-Abelian groups of orders 39 and 55.

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

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Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: Explain why a non-Abelian group of order 8 cannot be the internaldirect product of proper subgroups

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Q: Show that there are two Abelian groups of order 108 that haveexactly one subgroup of order 3.

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Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

Q: Find a non-trivial, proper normal subgroup of the dihedral group Dn-

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Q: Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal…

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Q: Prove that every group of order 675 has a normal Sylow 5-subgroup.

A: Theorem: If number of Sylow p-subgroup of a group G is unique, then that subgroup is normal. Let G…

Q: Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no subgroup…

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Q: Find a subgroup of Z12 ⨁ Z4 ⨁ Z15 that has order 9.

A: Given group is Z12⊕Z4⊕Z15. It is known that for each divisors r of n, Zn has exactly one cyclic…

Q: think of this as being a stronger type of normality. Prove that a characteristic subgroup is normal…

A: A subgroup H of h is called normal subgroup of h if θH⊆H  ∀θ∈AutG

Q: Prove that a group of order 595 has a normal Sylow 17-subgroup.

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Q: Prove that every group of order 78 has a normal subgroup of order 39.

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Q: How do you interprete the main theorem of Galois Thoery in terms of subgroup and subfield diagrams?

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Q: Prove that every group of order 375 has a subgroup of order 15.

A: According to the given information it is required to prove that every group of order 375 has a…

Q: Suppose that H is a subgroup of Sn of odd order. Prove that H is asubgroup of An.

A: Given: H is a subgroup of Sn of odd order, To prove: H is a subgroup of An,

Q: Show that a group of order 77 is cyclic.

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Q: Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic…

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