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Many problems on group theory

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Group Theory

Group Theory
1.
i. State the axioms for an equivalence relation
ii. The relation n mod 3 divides the non-negative integers
(i.e, n in Z such that n ≥ 0) into how many partitions?
Show that n = 0 mod 3 is an equivalence relation.

2. Prove that, for any matrices, A, B and C:
A+B=B+A
And:
A+(B+C)=(A+B)+C

( i.e., that the matrix addition is both commutative and associative)
For simplicity, prove these properties using 2x2 matrices.

3. Prove that addition modulo n, written + is:
i. Associative.
ii. Commutative.

There are two ways to prove these properties. Each way requires a definition or two:
i. For n ≥ 2, 0 ≤ a, b ≤ n+1,
a+ b= a+b if a+b< n
a+n-n if a+b&#8805; n
ii. Writing a for a mod n and (a+b) = a+ b, then:
(p+ q) &#8801; (p +q )

Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)?

4. Prove that addition modulo n, written + is:
i. Associative
ii Commutative.

( extra definations required : a for a mod n and (pà?q) = pà? q, so
(pà? q) (p à?q )
5.
i. State the axioms defining a group
- If (Z, +) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.
- If (Z, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z, identify the inverse. Also show that + is associative.
- If (R, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.

iii. In each case, deterimine whether the algebraic structure is a group. For each such group:
- show how it satifies the group axioms
- Draw the cayley table for the group and list the inverse elements

i. For S=(0,2), a+2b &#8801; (a+b) mod 2 and aà?2b &#8801; (a à?b)mod2
a. (S,+ ) ( possibly an additive group)
b. (S,&#8729; 2) (possibly a multiplicative group).

ii. For S = (0,1,2) where n=2,3 and + and à? are defined as in the last part.
a. (S,+n) ( possibly an additive group)
b. (S,&#8729; n) (possibly a multiplicative group).

Determine whether any of the groups is an abelian group. If any of them are abelian:
i. state the conditions under which a group is abelian
ii. show that the group is abelian

6. there are only two groups of order four (Z 4and v). How many groups are there of the order five? Draw cayley tables for each one of them( the element should be named a,b,c,d,e) is either (or both) of the groups of order four subgroup of any of those of order five? If so which one

7. for each of the following structures, state whethere it is a group. If it is, state whether it is abelian or not.

i.For any set, A, the set of one-to one and onto functions, f: A &#8594;A under composition ( written "&#9702;").

ii.The set of all subsets of the three-element set (a,b,c) ( there are eight such subsets) under:
a. Union
b. Intersection

iii. The set G=(a+b&#8730;5| a,b in Q) under addition and multiplication

iv The set consisting of non-zero numbers under
a. addition
b. division

v. The set (1,5,7,11) under multiplication modulo 12. Draw cayley table

vi. The set (4,6) under multiplication modulo 12. draw cayley table

vii. The set of real numbers under à?, where aà?b = 2(a+b)

viii The set of real numbers under +, where a+b = a+b-10

ix. The set of rotational symmetries of a regular hexagon under composition

x The following sets of permutations under composition

i. (e,(12),(123),(1234))
ii. (e,(12), (34), (12), (34))

8.Let G be a group, (G, * ) in which there is an element, a , such that g * g=g . prove that g=e

9.Prove that for every element, a, of a group, G, the order of a and a^-1 are the same ( including the case of an infinite order)

10.Let x and y be elements of a group, G. Prove that the elements xy and yx have the same orders

11.Find the subgroups of
i. Z7
ii. Z8
iii Z9
12.
i. determine which of the folowign are subgroups of under +
a. (0)
b. (-1,0,1)
c. (n| n=10m for some integer m
d.(p| p is a prime number
e. (0,1,2,3,4) under addition modulo 5
ii. Determine which of the following are subgroups of under mulitiplication:
a. (1, -1)
b. (x |x=3, for some integer n
c. (x |x=p/2&#8319; for some integers, p,n)
d. (x| x=k 3 for some interger k

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Solution Preview

Group Theory

Group Theory
1.
i. State the axioms for an equivalence relation
ii. The relation n mod 3 divides the non-negative integers
(i.e, n in Z such that n &#8805; 0) into how many partitions?
Show that n = 0 mod 3 is an equivalence relation.

2. Prove that, for any matrices, A, B and C:
A+B=B+A
And:
A+(B+C)=(A+B)+C

( i.e., that the matrix addition is both commutative and associative)
For simplicity, prove these properties using 2x2 matrices.

3. Prove that addition modulo n, written + is:
i. Associative.
ii. Commutative.

There are two ways to prove these properties. Each way requires a definition or two:
i. For n &#8805; 2, 0 &#8804; a, b &#8804; n+1,
a+ b= a+b if a+b< n
a+n-n if a+b&#8805; n
ii. Writing a for a mod n and (a+b) = a+ b, then:
(p+ q) &#8801; (p +q )

Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)?

4. Prove that addition modulo n, written + is:
...

Solution Summary

This solution is comprised of a detailed explanation of the problems of the group theory.
It contains step-by-step explanation for the following problem:

Group Theory
1.
i. State the axioms for an equivalence relation
ii. The relation n mod 3 divides the non-negative integers
(i.e, n in Z such that n &#8805; 0) into how many partitions?
Show that n = 0 mod 3 is an equivalence relation.

2. Prove that, for any matrices, A, B and C:
A+B=B+A
And:
A+(B+C)=(A+B)+C

( i.e., that the matrix addition is both commutative and associative)
For simplicity, prove these properties using 2x2 matrices.

3. Prove that addition modulo n, written + is:
i. Associative.
ii. Commutative.

There are two ways to prove these properties. Each way requires a definition or two:
i. For n &#8805; 2, 0 &#8804; a, b &#8804; n+1,
a+ b= a+b if a+b< n
a+n-n if a+b&#8805; n
ii. Writing a for a mod n and (a+b) = a+ b, then:
(p+ q) &#8801; (p +q )

Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)?

4. Prove that addition modulo n, written + is:
i. Associative
ii Commutative.

( extra definations required : a for a mod n and (pà?q) = pà? q, so
(pà? q) (p à?q )
5.
i. State the axioms defining a group
- If (Z, +) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.
- If (Z, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z, identify the inverse. Also show that + is associative.
- If (R, à?) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.

iii. In each case, deterimine whether the algebraic structure is a group. For each such group:
- show how it satifies the group axioms
- Draw the cayley table for the group and list the inverse elements

i. For S=(0,2), a+2b &#8801; (a+b) mod 2 and aà?2b &#8801; (a à?b)mod2
a. (S,+ ) ( possibly an additive group)
b. (S,&#8729; 2) (possibly a multiplicative group).

ii. For S = (0,1,2) where n=2,3 and + and à? are defined as in the last part.
a. (S,+n) ( possibly an additive group)
b. (S,&#8729; n) (possibly a multiplicative group).

Determine whether any of the groups is an abelian group. If any of them are abelian:
i. state the conditions under which a group is abelian
ii. show that the group is abelian

6. there are only two groups of order four (Z 4and v). How many groups are there of the order five? Draw cayley tables for each one of them( the element should be named a,b,c,d,e) is either (or both) of the groups of order four subgroup of any of those of order five? If so which one

7. for each of the following structures, state whethere it is a group. If it is, state whether it is abelian or not.

i.For any set, A, the set of one-to one and onto functions, f: A &#8594;A under composition ( written "&#9702;").

ii.The set of all subsets of the three-element set (a,b,c) ( there are eight such subsets) under:
a. Union
b. Intersection

iii. The set G=(a+b&#8730;5| a,b in Q) under addition and multiplication

iv The set consisting of non-zero numbers under
a. addition
b. division

v. The set (1,5,7,11) under multiplication modulo 12. Draw cayley table

vi. The set (4,6) under multiplication modulo 12. draw cayley table

vii. The set of real numbers under à?, where aà?b = 2(a+b)

viii The set of real numbers under +, where a+b = a+b-10

ix. The set of rotational symmetries of a regular hexagon under composition

x The following sets of permutations under composition

i. (e,(12),(123),(1234))
ii. (e,(12), (34), (12), (34))

8.Let G be a group, (G, * ) in which there is an element, a , such that g * g=g . prove that g=e

9.Prove that for every element, a, of a group, G, the order of a and a^-1 are the same ( including the case of an infinite order)

10.Let x and y be elements of a group, G. Prove that the elements xy and yx have the same orders

11.Find the subgroups of
i. Z7
ii. Z8
iii Z9
12.
i. determine which of the folowign are subgroups of under +
a. (0)
b. (-1,0,1)
c. (n| n=10m for some integer m
d.(p| p is a prime number
e. (0,1,2,3,4) under addition modulo 5
ii. Determine which of the following are subgroups of under mulitiplication:
a. (1, -1)
b. (x |x=3, for some integer n
c. (x |x=p/2&#8319; for some integers, p,n)
d. (x| x=k 3 for some interger k

Solution contains detailed step-by-step explanation.

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14. Evolutionary theories often emphasize that humans have adapted to their physical environment. One such theory hypothesizes that people should spontaneously follow a 24-hour cycle of sleeping and waking-even if they are not exposed to the usual pattern of sunlight. To test this notion, eight paid volunteers were placed (individually) in a room in which there was no light from the outside and no clocks or other indications of time. They could turn the lights on and off as they wished. After a month in the room, each individual tended to develop a steady cycle. Their cycles at the end of the study were as follows:
25, 27, 25, 23, 24, 25, 26, and 25.

Using the .05 level of significance, what should we conclude about the theory that 24 hours is the natural cycle? (That is, does the average cycle length under these conditions differ significantly from 24 hours?)

(a) Use the steps of hypothesis testing.
(b) Sketch the distributions involved.
(c) Explain your answer to someone who has never taken a course in statistics.

18. Twenty students randomly assigned to an experimental group receive an instructional program; 30 in a control group do not. After 6 months, both groups are tested on their knowledge. The experimental group has a mean of 38 on the
test (with an estimated population standard deviation of 3); the control group has a mean of 35 (with an estimated population standard deviation of 5). Using the .05 level, what should the experimenter conclude?

(a) Use the steps of hypothesis testing,
(b) sketch the distributions involved, and
(c) explain your answer to someone who is familiar with the t test for a single sample but not with the t test for independent means.

17. Do students at various universities differ in how sociable they are? Twenty-five students were randomly selected from each of three universities in a region and were asked to report on the amount of time they spent socializing each day with
other students. The result for University X was a mean of 5 hours and an estimated population variance of 2 hours; for University Y, M = 4, S2 = 1.5 ; and for University Z, . M = 6, S2 = 2.5 What should you conclude? Use the .05 level.

(a) Use the steps of hypothesis testing,
(b) figure the effect size for the study; and
(c) explain your answers to parts (a) and (b) to someone who has never had a course in statistics.

11. Make up a scatter diagram with 10 dots for each of the following situations:

(a) perfect positive linear correlation,
(b) large but not perfect positive linear correlation,
(c) small positive linear correlation,
(d) large but not perfect negative linear correlation,
(e) no correlation,
(f) clear curvilinear correlation.

For problem 12 do the following:
(a) Make a scatter diagram of the scores;
(b) describe in words the general pattern of correlation, if any;
(c) figure the correlation coefficient;
(d) figure whether the correlation is statistically significant (use the .05 significance level, two-tailed);
(e) explain the logic of what you have done, writing as if you are speaking to someone who has never heard of correlation (but who does understand the mean, deviation scores, and hypothesis testing); and
(f) give three logically possible directions of causality, indicating for each direction whether it is a reasonable explanation for the correlation in light of the variables involved (and why).

12. Four research participants take a test of manual dexterity (high scores mean better dexterity) and an anxiety test (high scores mean more anxiety). The scores are as follows.

Person Dexterity Anxiety
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4 4 -2

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