Contemporary Abstract Algebra

9th Edition

ISBN: 9781337249560

Author: Joseph Gallian

Publisher: Cengage Learning US

See similar textbooks

Videos

Textbook Question

Chapter 3, Problem 78E

Let G be a finite group with more than one element. Show that G has an element of prime order.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3, Problem 77E

Chapter 4, Problem 1E

Students have asked these similar questions

4. Solve the system of equations and express your solution using vectors. 2x1 +5x2+x3 + 3x4 = 9 -x2+x3 + x4 = 1 -x1-6x2+3x3 + 2x4 = -1

3. Simplify the matrix expression A(A-B) - (A+B)B-2(A - B)2 + (A + B) 2

[2 pts] 1. Let A = [. 1 -1 0 -343 and B = 05 5 -7 304 Compute (7A - 3B) - 4(2A - B).

Chapter 3 Solutions

Contemporary Abstract Algebra

Ch. 3 - For each group in the following list, find the...Ch. 3 - Let Q be the group of rational numbers under...Ch. 3 - Let Q and Q* be as in Exercise 2. Find the order...Ch. 3 - Prove that in any group, an element and its...Ch. 3 - Without actually computing the orders, explain why...Ch. 3 - In the group Z12 , find a,b,anda+b for each case....Ch. 3 - If a, b, and c are group elements and a=6,b=7 ,...Ch. 3 - What can you say about a subgroup of D3 that...Ch. 3 - What can you say about a subgroup of D4 that...Ch. 3 - How many subgroups of order 4 does D4 have?

Ch. 3 - Determine all elements of finite order in R*, the...Ch. 3 - Complete the statement “A group element x is its...Ch. 3 - For any group elements a and x, prove that xax1=a...Ch. 3 - Prove that if a is the only element of order 2 in...Ch. 3 - (1969 Putnam Competition) Prove that no group is...Ch. 3 - Let G be the group of symmetries of a circle and R...Ch. 3 - For each divisor k1 of n, let Uk(n)=xU(n)xmodk=1...Ch. 3 - Suppose that a is a group element and a6=e . What...Ch. 3 - If a is a group element and a has infinite order,...Ch. 3 - For any group elements a and b, prove that ab=ba .Ch. 3 - Show that if a is an element of a group G, then...Ch. 3 - Show that U(14)=3=5 . [Hence, U(14) is cyclic.] Is...Ch. 3 - Show that U(20)k for any k in U(20). [Hence, U(20)...Ch. 3 - Suppose n is an even positive integer and H is a...Ch. 3 - Let n be a positive even integer and let H be a...Ch. 3 - Prove that for every subgroup of Dn , either every...Ch. 3 - Let H be a subgroup of Dn of odd order. Prove that...Ch. 3 - Prove that a group with two elements of order 2...Ch. 3 - Prob. 29E Ch. 3 - Prob. 30E Ch. 3 - Prob. 31E Ch. 3 - Suppose that H is a subgroup of Z under addition...Ch. 3 - Prove that the dihedral group of order 6 does not...Ch. 3 - If H and K are subgroups of G, show that HK is a...Ch. 3 - Let G be a group. Show that Z(G)=aGC(a) . [This...Ch. 3 - Let G be a group, and let aG . Prove that...Ch. 3 - For any group element a and any integer k, show...Ch. 3 - Let G be an Abelian group and H=xG||x is odd}....Ch. 3 - Prob. 39E Ch. 3 - Prob. 40E Ch. 3 - Let Sbe a subset of a group and let H be the...Ch. 3 - In the group Z, find a. 8,14 ; b. 8,13 ; c. 6,15 ;...Ch. 3 - Prove Theorem 3.6. Theorem 3.6 C(a) Is a Subgroup...Ch. 3 - If H is a subgroup of G, then by the centralizer...Ch. 3 - Must the centralizer of an element of a group be...Ch. 3 - Suppose a belongs to a group and a=5 . Prove that...Ch. 3 - Prob. 47E Ch. 3 - In each case, find elements a and b from a group...Ch. 3 - Prove that a group of even order must have an odd...Ch. 3 - Consider the elements A=[0110]andB=[0111] from...Ch. 3 - Prob. 51E Ch. 3 - Give an example of elements a and b from a group...Ch. 3 - Consider the element A=[1101] in SL(2,R) . What is...Ch. 3 - For any positive integer n and any angle , show...Ch. 3 - Prob. 55E Ch. 3 - In the group R* find elements a and b such that...Ch. 3 - Prob. 57E Ch. 3 - Prob. 58E Ch. 3 - Prob. 59E Ch. 3 - Compute the orders of the following groups. a....Ch. 3 - Let R* be the group of nonzero real numbers under...Ch. 3 - Compute U(4),U(10),andU(40) . Do these groups...Ch. 3 - Find a noncyclic subgroup of order 4 in U(40).Ch. 3 - Prove that a group of even order must have an...Ch. 3 - Let G={[abcd]|a,b,c,dZ} under addition. Let...Ch. 3 - Let H=AGL(2,R)detA is an integer power of 2}. Show...Ch. 3 - Let H be a subgroup of R under addition. Let...Ch. 3 - Let G be a group of functions from R to R*, where...Ch. 3 - Let G=GL(2,R) and...Ch. 3 - Let H=a+bia,bR,ab0 . Prove or disprove that H is...Ch. 3 - Let H=a+bia,bR,a2+b2=1 . Prove or disprove that H...Ch. 3 - Let G be a finite Abelian group and let a and b...Ch. 3 - Prob. 73E Ch. 3 - If H and K are nontrivial subgroups of the...Ch. 3 - Prob. 75E Ch. 3 - Prove that a group of order n greater than 2...Ch. 3 - Let a belong to a group and a=m. If n is...Ch. 3 - Let G be a finite group with more than one...

Knowledge Booster

$Algebra$

Learn more about

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