
Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
9th Edition
ISBN: 9781305657977
Author: Gallian, Joseph
Publisher: Brooks Cole
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Chapter 3, Problem 75E
To determine
To prove: H has a nontrivial proper subgroup.
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Kate, Luke, Mary and Nancy are sharing a cake. The cake had previously been divided into four slices (s1, s2, s3 and s4). The following table shows the values of the slices in the eyes of each player. What is fair share to nancy?
S1
S2
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S4
Kate
$4.00
$6.00
$6.00
$4.00
Luke
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$5.00
$5.25
$5.45
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$4.50
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Kate, Luke, Mary and Nancy are sharing a cake. The cake had previously been divided into four slices (s1, s2, s3 and s4). The following table shows the values of the slices in the eyes of each player.
S1
S2
S3
S4
Kate
$4.00
$6.00
$6.00
$4.00
Luke
$5.30
$5.00
$5.25
$5.45
Mary
$4.25
$4.50
$3.50
$3.75
Nancy
$6.00
$4.00
$4.00
$6.00
how much is the cak worth to mary
Kate, Luke, Mary and Nancy are sharing a cake. The cake had previously been divided into four slices (s1, s2, s3 and s4). The following table shows the values of the slices in the eyes of each player. What is the threshold of fair share for Luke?
S1
S2
S3
S4
Kate
$4.00
$6.00
$6.00
$4.00
Luke
$5.30
$5.00
$5.25
$5.45
Mary
$4.25
$4.50
$3.50
$3.75
Nancy
$6.00
$4.00
$4.00
$6.00
Chapter 3 Solutions
Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
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. 29ECh. 3 - Prob. 30ECh. 3 - Prob. 31ECh. 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. 39ECh. 3 - Prob. 40ECh. 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. 47ECh. 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. 51ECh. 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. 55ECh. 3 - In the group R* find elements a and b such that...Ch. 3 - Prob. 57ECh. 3 - Prob. 58ECh. 3 - Prob. 59ECh. 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. 73ECh. 3 - If H and K are nontrivial subgroups of the...Ch. 3 - Prob. 75ECh. 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...
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