Let G be a finite group of order 125 with the identity element e and assume that G contains an element a with a²5 ± e Prove that G is cyclic.

Q: Let G be a group of order pm where p is a prime number and m is a positive integer. Show that G…

A: Let G be a group of order pm, where p is a prime number and m is a positive integer. Then,…

Q: M  be a group (not necessarily an Abelian group) of order 387. Prove that M  must have an element of…

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Q: Prove that, if H is a subgroup of a cyclic group G, then the quotient group G/H is also cyclic.

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Q: Let G be a finite group of order n. Let g be an element of G. Prove that gn=e.

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Q: Let G be the subgroup of GL3(Z₂) defined by the set 100 a 10 bc1 such that a, b, c Z₂. Show that G…

A: The given set of matrix is 100a10bc1 where a, b, c∈ℤ2. To find: the group to which the given set is…

Q: Let N be a normal subgroup of a finite group G. Use the theorems ofthis chapter to prove that 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: Let G be a finite cyclic group of order 20, and a in G. Then one of the following is possible order…

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Q: let G be an abelian Group and let H and K be a susb group of G prove that HK ={hk / h an element H…

A: Consider the provided question, Your question is missing the last statement; I think you want to…

Q: Let G be a finite group of order n and identity element 1. a. Show that g" =1 for any g e G. b. An…

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Q: Suppose that G is a finite group with the property that every nonidentityelement has prime order…

A: This can be proved by lemma that states If G is abelian with the property that every nonidentity…

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

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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: Let G be a finite group of order n, positive integer, with identity e. Prove or disprove: For any g…

A: Let G be a finite group of order n, with identity e i.e, order of (G) = |G| = n .   Now let g be any…

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: 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: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

Q: Let G be a finite group of order p  where p is prime. Prove that G is cyclic.

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

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Q: Let l be an integer greater than 1, and let G be a finite group with no element of order l. Can…

A: Find the attachment for the solution.

Q: Let G be a group with |G|=187 then every proper subgroup of G is: * Non cyclic None of these Сyclic…

A: Given that G is a group and G=187 i.e., number of elements in G is 187.  First we have to find the…

Q: Let G be a finite group. Let x E G and let G be the conjugacy class of r. Prove that |#C| < |[G,…

A: Given G be a finite group. Let x∈G. Define: xG=gxg-1: g∈G Let G:G=m. Define a map π:G→GG:G defined…

Q: Let G= be a cyclic group of order 60, and H=. List all left cosets of H in G.

A: Let G be a group and H be a subgroup of G. We define the left co-set of H by a as aH = ah : h∈H Then…

Q: Let G, H be cyclic groups of order 2 and 3, respectively. Find the orders of all elements of G x H

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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: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…

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Q: Let H be the set of all elements of the abelian group G that have finite order. Prove that H is a…

A: Let H be the set of all elements of the abelian group G that have finite order.  Prove that H is a…

Q: Let G be a group of finite order n. Prove that an = e for all a in G.

A: Let G be a group of finite order n with identity e.           Since G is of finite order…

Q: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

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Q: Let G be a cyclic group of order 33. List the orders of all its elements and how many elements have…

A: Here, G is a cyclic group of order 33.  We have to find the orders of all its elements and the…

Q: prove that a group G of order p^2, where p is a prime, is abelian.

A: Suppose, G is a group of order p2 where p is prime.

Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

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Q: Let G be a group of order pq where p and q are distinct primes andp &lt; q. Prove that the Sylow…

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Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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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: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

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Q: Let G be a group with |G|=187 then every proper subgroup of G is: * Non abelian Non cyclic None of…

A: See the detailed solution below.

Q: Let G be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

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: Let G be a group with |G|=221 then every proper subgroup of G is: ONone of these Non abelian Non…

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Q: Prove that the order of the group element gN in G/N divides the order of g.

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Q: Let G be a group with no proper, nontrivial subgroups and assume that |G| > 1. Prove that G must be…

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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: Let G be a finite non-abelian simple group and let q be prime, then [G] is

A: It is given that G be any finite non Abelian simple group and q be any prime. We have to determine…

Q: 6. Let G be a group of order p², where p is a prime. Show that G must have a subgroup of order p.

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Q: Let G be a group of order 24. Suppose that G has precisely one subgroup of order 3, and one subgroup…

A: Theorem : If a group G is the internal direct product of subgroups H and K, then G is isomorphic to…