Let G be a group of order 25. Prove G is cyclic or g^5=e for all g in G. Generalize to any group of order p^2, where p is prime. Does proof work for this generalization? I am stuck on the generalizing part.

Let G be a group of order 25. Prove G is cyclic or g^5=e for all g in G. Generalize to any group of order p^2, where p is prime. Does proof work for this generalization? I am stuck on the generalizing part.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 4E: 4. Prove that the special linear group is a normal subgroup of the general linear group .

See similar textbooks

Related questions

Q: Suppose that G is a finite abelian group. Prove that G has order pn where p is prime, if and only…

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: 6. Give an example of two groups with 9 elements each which are not isomorphic to each other (and…

Q: Prove or Disprove: If (G, *) be an abelian group, then (G, *) a cyclic group?

A: If the given statement is true then we will proof the statement otherwise disprove we taking the…

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: Show that the group of permutations Σ2 is abelian. Then show that Σ3 is not. Writing up the group…

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

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

A: consider the equation

Q: give an example of a finite, non-cyclic abelian group containing a container of order 5

A: Take the abelian group G=Z5×Z5 of order 25 whose every element (except identity) is of order 5 and…

Q: A group that also satisfies the commutative property is called a(n). (or abelian) group. A group…

A: A group that also satisfies the commutative property is called a(n) commutative group (or abelian)…

Q: Find Aut(Z15) . Use the Fundamental Theorem of Abelian Groups to express this group as an external…

A: We know the following theorem. Theorem: Aut(Zn) is isomorphic to the group of multiplicative units…

Q: Find any case in which the number of subgroups with an order of 3 can be exactly 4 in the Abelian…

A: Let G be an abelian group of order 108 Find the number of subgroups of order 3. Prove that, in any…

Q: Use the theory of finite abelian groups to find all abelian groups of order 4. Explain why there are…

A: we know that if G is finite group whose order is power of a prime p thenZ(G) has more than one…

Q: Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12…

Q: Give an example of elements a and b from a group such that a hasfinite order, b has infinite order…

Q: How many subgroups of a not abelian group of order 6 is non-cyclic? Select one:

A: Given: The order of the group = 6.

Q: (1) State the structure theorem for finite generated Abelian group. Using this theorem to classify…

A: To find the all finite generated Abelian group of order 120: Structure theorem for finite generated…

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

Q: Determine all cyclic groups that have exactly two generators.

Q: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

Q: (1) Z/12Z

Q: Show that a subgroup of a solvable group is solvable.

Q: Without using the structure theorem for finite abelian groups, prove that a finite abelian group has…

A: Given: Let G be a finite abelian group. To prove: G has an element of order p, for every prime…

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: Let G be a group (not ncesssarily an Abelian group) of order 425. Prove that G must have an element…

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: Show that every abelian group of order 255 (3)(5)(17) is isomorphic to Z55 and hence cyclic. [Ilint:…

A: We have to solve given problem:

Q: Which abelian somorphic to groups subyraups of Sc. Explin. are

A: Writing a permutation σ∈Sn as a product of n disjoint circles. i.e σ=τ1,τ2,τ3,…τk The order of σ is…

Q: Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.

Q: ake the minimum three groups along with subgroups (Mathematically) and verify the langrage theorem

A: Lagrangian theorem says that, If H is a subgroup of a group G then H\G. Now take, Z3={0,1,2} The…

Q: Prove that

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

Q: Find Aut(Z20). Use the fundamental theorem of Abelian groups to express this group as an external…

A: Find Aut(Z20) by using the fundamental theorem of Abelian groups

Q: ) Describe, up to isomorphism, all abelian groups of order 300 = 22 . 3 - 52 using the Fundamental…

A: Solution: Given : Group of order 300

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: List six examples of non-Abelian groups of order 24.

A: The Oder is 24

Q: Show that the groups Z8xZ20xZ12 and Z120xZ4xZ4 are isomorphic by define a one-one and onto map? what…

A: We will use the basic knowledge of groups and abstract algebra to answer this question.

Q: Show that a homomorphism defined on a cyclic group is completelydetermined by its action on a…

A: Consider the x is the generator of cyclic group H for xn∈H, ∅(x)=y As a result, For all members of…

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

Q: Show that there exists a non trivial normal subgroup of a group of order pq, where p and q are prime…

A: It is given that G is a group order pq. To prove G has a non-trivial normal subgroup. It is known…

Q: Show that there are two Abelian groups of order 108 that haveexactly 13 subgroups of order 3.

A: Aim: There are two Abelian groups of order 108 that have exactly 13 subgroups of order 3.

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

Q: 300Can someone please help me understand the following problem. I need to know how to start the…

A: G is the abelion group of order 16. It is isomorphic to,

Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering second…

Q: Check whether a group of order 156 is simple or not.

A: Definition: A group G is said to be simple if the only normal subgroups of G are the trivial group e…

Q: t subgroups and quotient groups of a solvable group are solvable.

Q: (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a…

A: Given that G is a group of order pq, where p and q are distinct prime numbers and G has only one…

Q: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

A: Two groups G and G' are isomorphic i.e., G≃G′, if there exists an isomorphism from G to G'. In…

Q: 15*. Find an explicit epimorphism from Z24 onto a group of order 6. (In your work, identify the…

A: To construct a homomorphism from Z24 , which is onto a group of order 6.

Q: Let F denote the set of first 8 fibonacci number.Then convert into a non.abelian group by showing…

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

Question

Let G be a group of order 25. Prove G is cyclic or g^5=e for all g in G. Generalize to any group of order p^2, where p is prime. Does proof work for this generalization?

I am stuck on the generalizing part.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Answer

Trending now

This is a popular solution!

Step by step

Solved in 7 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Find two groups of order 6 that are not isomorphic.
4. Prove that the special linear group is a normal subgroup of the general linear group .
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
Show that every subgroup of an abelian group is normal.
Find all subgroups of the quaternion group.
Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
4. List all the elements of the subgroupin the group under addition, and state its order.
Find all homomorphic images of the quaternion group.
3. Consider the group under addition. List all the elements of the subgroup, and state its order.
Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.
10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .
5. Exercise of section shows that is a group under multiplication. a. List the elements of the subgroupof , and state its order. b. List the elements of the subgroupof , and state its order. Exercise 33 of section 3.1. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and is designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.