Let G be a group of order pm where p is a prime number and m is a positive integer. Show that G contains an element of order p.

Q: Let G be a group and a E G be a certain fixed element of G. The centralizer of a in G is C(a) = {g €…

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: Let (G,*) be an a belian group, if (H,) and (K,*) are subgroup of (G,*) then (H * K,*) is a subgroup…

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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: 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 finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the…

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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 a group. Show that for all a, b E G, (ab)2 = a2b2 G is abelian

A: To prove that the group G is commutative (abelian) under the given conditions

Q: If a is an element of order 8 of a group G, and

A: Let G be a group. Let a is an element of order 8 of group G. That is, a8=e          where e is an…

Q: Let N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that the…

A: According to the given information, Let N be a finite group and H be a subgroup of N. H is odd and…

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 S, be the symmetric group and let a be an element of S, defined by: 1 2 3 4 5 67 8 9 ) B = (7…

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Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

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 group and a ∈ G. The centralizer of  a in G is equal to the centralizer of a^-1 in G.…

A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…

Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…

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Q: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.

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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: If G is a finite group with |Gl<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Given: The group G is a finite group with | G | &lt; 120 and G has subgroups of orders 10, 15, and…

Q: Given the groups R∗ and Z, let G = R∗ ×Z. Define a binary operation ◦ on G by (a, m) ◦ (b, n) = (ab,…

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Q: Let C be a group with |C| = 44. Prove that Cmust contain an element of order 2.

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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: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

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 order 24. If H is a subgroup of G, what are all the possible orders of H?

A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…

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 a group of order 100. If G (a) The generators of Gare: a.a.a a to 30 45 60 75 90 15 35 45…

A: Theorem Used: Let G=&lt;g&gt; be a cyclic group where O(G)=n. Then G=&lt;gk &gt; if and only if gcd…

Q: Let M be an abelian group and D a subgroup. For any element m of M, where m has order 2, define…

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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: * Let H be a proper subgroup of the group (Z/n)*. Prove that there are infinitely many prime numbers…

A: Given H is subgroup of Z/n* Assume there exist q1, q2 with 0≤q1,q2≤1 and q1≠q2 such that q1+Z=q2+Z…

Q: Let G be a group with p or more distinct elements of order p for some prime p. Show that G is not…

A: Given : Let G be a group which has p  elements of order p, p is a prime no.  To show G is not cyclic…

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: Let G be a group of order 425. Prove that if H is a normal bgroup of order 17 in G then H < Z(G).

A: Let G be a group of order 425. Prove that if H is a normal subgroup of order 17 in G. Then H≤ Z(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: 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: 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 any group of order 75 can have at most one subgroup of order 25.

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

A: given G be a group of order pm where p is a prime number and m is a positive integer to show G…

Q: Let (G, ) be a group. Define a new binary operation * on G by the formula a * b = b - a for all a…

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Q: Let G be an abelian group, then (acba)(abc)¯1 is

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Q: G is a finite group of order IGI =pqr with p< q <r prime

A: Solution

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: 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: Consider the alternating group A4. Identify the groups N and A4 /N up to an isomorphism.

A: Consider the alternating group A4. We need to Identify the groups N and A4 /N up to an isomorphism.…

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…

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.