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

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

Q: Prove that a simple group of order 60 has a subgroup of order 6 anda subgroup of order 10.

A: If G is the simple group of order 60 That is | G | =60. |G| = 22 (3)(5). By using theorem, For every…

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

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Q: Prove that any group with prime order is cyclic.

A: Given, Any group with prime order. let o(G)=p (p is a prime number) we assure that G has no subgroup…

Q: Let (G,*) be a group of order p, q, where p, q are primes and p < q. Prove that (a). G has only one…

A: It is given that G, * is a group of order p·q where p, q are primes and p&lt;q. Show that G has only…

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

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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: All groups of order three are isomorphic.

A: All groups of order three are isomorphic.

Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * True False

A: let H be a cyclic subgroup of a group G.

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

A: Given the order of the group is 3, we have to prove this is a cyclic group.

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: Let G be a group of order 100 that has a subgroup H of order 25.Prove that every element of G of…

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

A: Solution: The objective is to find a proper subgroup of the group of integers and to show that this…

Q: Let G be the subgroup of GL3(Z2) defined by the set 100 a 10 b C 1 that a, b, c Z₂. Show that G is…

A: Given: G is the subgroup of GL3ℤ2 which is defined by the set of matrix 100a10bc1 where a, b, c∈ℤ2 .…

Q: Prove that there is no simple group of order 280 = 23 .5 . 7.

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Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

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Q: Let Z denote the group of integers under addition. Is every subgroup of Z cyclic? Why? Describe all…

A: Yes , every subgroup of z is cyclic

Q: Prove that every subgroup of an abelian group is abelian

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Q: Prove that A5 is the only subgroup of S5 of order 60.

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Q: Let H  and K  be subgroups of a group G with operation * .  Prove that HK  .is closed  under the…

A: Given information: H  and K  be subgroups of a group G with operation * To prove that HK is a closed…

Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

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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: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

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Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

Q: Let G be a group with the order of G = pq, where p and q are prime. Prove that every proper subgroup…

A: Consider the provided question, Let G be a group with the order of G = pq, where p and q are prime.…

Q: Prove that there is no simple group of order 210 = 2 . 3 . 5 . 7.

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Q: Prove that any two Sylow p-subgroups of a group G are conjugate.

A: To prove that any two sylowp-subgroup of a group G are conjugate

Q: Let G be a group of order 90. show that G has at most one subgroup of order 45

A: Given: G be a group of order 90

Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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Q: Suppose that G is a group of order 105 with the property that G hasexactly one subgroup for each…

A: Given G is a group of order 105 .

Q: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

Q: Let let G N Subgroup be be of G a a group and normal of finite

A: To prove that H is contained in N, we first prove this: Lemma: Let G be a group.H⊂G. Suppose, x be…

Q: Prove that a group that has more than one subgroup of order 5 musthave order at least 25.

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Q: Prove that the intersection of two subgroups is always a subgroup.

A: In this question, we prove the intersection of the two subgroup of G is also the subgroup of G.

Q: Let G be a group, and N ⊆ Z(G) be a subgroup of the center of G, Z(G). If G/N, the quotient group is…

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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: Any group of order 520 is simple

A: Simple group:     "A simple group is a nontrivial group whose only normal subgroups are the trivial…

Q: Prove that if B is a subgroup of G then the coset produced by multiplying every element of B with X…

A: Solution: Let us consider (G, .) be a group and B be a subgroup of the group G. Now for any g∈G, the…

Q: Let G be a finite group of order 125 with the identity element e and assume that G contains an…

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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: Prove that the intersection of two subgroups of G is itself a subgroup of G.

A: Suppose that, H1 and H2 be the sub groups of G. Also, let H=H1∩H2 . Consider one step group test:…

Q: Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G is cyclic.

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

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Q: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

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

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Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

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

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Q: Show that every group G of order n is isomorphic to a subgroup of Sn. (This is also called Caley's…

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