All references to specific exercises correspond to "Contemporary Abstract Algebra," 9th edition [8th edition in brackets, if different from 9th edition], Brooks Cole Cengage Learning, by Joseph A. Gallian

You should periodically reread this page as the course progresses since many of the comments refer to specific situations that will arise from time to time.

Unfortunately many students struggle with this subject. First and foremost you must study the material regularly. Don't wait until a day or two before an exam or when homework is due.

One of the greatest problems I see with students is that they do not have the
definitions and the theorems memorized. They will come to my office and say
that they can't do a certain problem such as prove some subset of a group is a subgroup. I will
say to them "What does the One Step Subgroup Test say?" They won't know. Or I
will ask them "What does |*a*| = 6 mean?" They will reply incorrectly that *a ^{6} = e*. Well, of
course, if you do not know the definitions and theorems you won't be able to use
them.

To learn to do proofs pick out several of the easier proofs that are given in the book (for instance, Example 5 of Chapter 3, Theorem 3.4 and Theorem 4.1). Write the statements down but not the proofs. Then see if you can prove them. Students often try to prove a statement without using the entire hypothesis. Keep in mind that you MUST use the hypothesis. If you cannot prove the statement, look at the first line of the proof in the text. That might be enough to get you started. If it is not, then look at the next line. Repeat this over and over until you can do them without looking at the text. Eventually you will get the hang of it. There is a direct relationship between your understanding of the subject and your ability to do proofs. Proofs test your understanding. They also test your creativity.

Keep in mind that you can only use what you have. For example, Exercise
28 [Exercise 26 in 8th ed] in Chapter 3 says that if you have an Abelian (that is, commutative) group with two elements
of order 2 then
it has a subgroup of order 4. So we can let *a* and *b* be the two elements of
order 2. Now all we have are* a* and * b* and the group axioms so USING ONLY *a* and *b*
you must create a subgroup of order 4. Well, the axioms tell us that the
identity is in the subgroup and closure tells us that *ab* is in there too so the
subgroup must be {*e,a,b, ab*}. Then all we need do is to show that *ab* is distinct from the other three elements and use
the Finite Subgroup Test to prove that
this set is a subgroup.

Keep in mind that any group theoretic argument must involve the group operation. Exercise 22 in Chapter 4 says to prove that a group of order 3 is cyclic.
In order to use the group operation you must give the elements names. So begin with
"Let *G* = {*e,a,b*}." Since groups are closed and all you have besides the identity *e* is *a* and *b* your argument MUST involve the term
*ab*. So look at *ab*. Because cancellation rules out
*ab = a* and *ab = b* we have that *ab = e*. Then *b = a*^{-1}. This proves that *G* is cyclic.

Here is another example. Look at Exercise 78 [80 in 8th edition] in Chapter 3 This exercise says to prove that a group with
more than one element must have an element of prime order. Since
the group has more than one element we may let *a* be a nonidentity
element. If *a* has prime order we are done. If *a* does not have prime order then
USING only *a* we must find an element of prime order. Since all we have to work with is *a*, the element of
prime order has to be found using *a*. So, consider *a, a ^{2}, a^{3},
a^{4}* etc. One of
these must have prime order. But how do we know which one? Well, try some
abstract examples (not specific examples such as

Another thing that you must do is learn in **WORDS** what each concept given in
symbols means. For example, *Z(G)* is the set of all elements that commute with
**EVERY** element of *G* while *C*(*a*) is the set of all elements that
commute with *a*. Likewise, you should think of |*a*| as the ** SMALLEST**
positive power of *a* that gives the identity.

Whenever possible, convert words to symbols. For example, if you are given that a group *G* is finite
write "Let |*G*| = *n*."
If you are given that a group is Abelian write "We know that * ab = ba * for all *a * and *b*."
If you are asked to prove that a group is Abelian write "We want to show that * ab = ba * for all *a * and *b*."
If you are given that a group has an element has order 10. Write
"We know that there is an element *a* such that *a ^{10} = e* and 10 is the least positive integer

Use suggestive notation. For example, if a problem involves one element that is fixed and one that varies or is unknown, denote the fixed element by

Whenever you are doing an exercise from the book or a problem on an exam you should ask yourself ``Is there a theorem in the book whose statement seems similar to the statement of the problem?" Most exercises and exams problems can be easily done by using one of the theorems in the book. For example, Exercise 25 in of Chapter 2 asks to prove a group is Abelian if and only if (

Here are some remarks about how to do algebra problems.

1. When you are asked to prove a statement you must not assume that the statement is true.

2. Never assume a group is Abelian. Some people begin their argument for Exercise 47 of Chapter 2 by saying "Assume that the group is Abelian." This is incorrect for you have no reason to assume a group is Abelian. Many groups are not Abelian.

3. Never divide group elements. Instead, use cancellation or inverses.

4. Never assume a group is finite when that condition was not stated.

5. In the text it is usually the case that elements of a group are
denoted by
letters from the beginning of the alphabet* a,b, c* or end of the
alphabet* x,y,z*.
Integers such as exponents and orders of elements or groups are usually denoted
with letters from the middle of the alphabet *i,j,k,m,n,s,t*. For
example, let
*|a| = n*. You should use the same conventions.

6. When asked to find the inverse of an element, always check your
answer by
multiplying the element and its purported inverse to see if you get the
identity. For example, to check that *(ab) ^{-1} = b^{-1}a^{-1}* all you need do is observe that

7. After you finish a proof look to see if you have used all the hypotheses. For example, if you were given that the group is Abelian check to see if you used that condition in your argument. If the hypothesis says the group is finite check to see where you used finiteness. Occasionally, it may be the case that a given condition is not really needed but was there just to make the problem easier but usually all the given conditions are needed for the you to be able to give a valid proof with what you know at this point in the book.

8. Many exercises in the book involve a parameter * n* and ask you to prove something.
(For example, Exercises 23, 27, and 28 in Chapter 2). You should look at the cases for small values of * n* such as 2 and 3 to gain insight and
look for a pattern. This often tells you how to do the general case but keep in mind that doing specific values for * n* does not do the general
case. The
problem must be done for all* n, * not a few
examples. In general, you cannot prove a statement is true by using specific examples.

9. When ask to provide an example to illustrate something, a dihedral group such as *D _{4}* is often a good
group to try. For example, Exercise 12 of Chapter 2.

10. On problems that ask for some answer rather that to prove something do
not just give an answer. Show that your answer is valid. You must give reasons
or an explanation of why your answer is correct. One example is
"If
*a _{1},a_{2},...,a_{n}*
belong to a group, what is the inverse of

11. In many cases problems can be solved by simply writing out the expressions.
For example in Exercise 34 of Chapter 2 write out *(ab) ^{2} = a^{2}
b^{2}* as

12. Many theorems in the book about groups and elements of groups involve divisibility conditions and greatest common divisors of two integers. Divisibility only applies to integers. Infinity is not an integer. Do not talk about an integer dividing infinity or an integer being relatively prime to infinity.

13. Whenever you say *"Assume ..."* you must have a reason why you may assume
what it is you are assuming. For example, if you are given that * H* is a subgroup
of *G* you may make the statement: Assume *x* is an element of *H* because subgroups
are not empty. You cannot say "Assume *G* is Abelian" without providing some
reason why you may assume that *G* is Abelian. As another example, if you are
given that a group is finite and *a* is an element of the group you may say
"Assume |*a*| = *n*" because all elements of a finite group have finite order.
However, if you do not know that the group is finite you can't assume that an
arbitrary element from the group has finite order. Instead, you should take two
cases. Case 1: *|a|* is finite and Case 2: *|a|* is
infinite.

14. When doing a problem about the order of an element, such as proving
that an element and its inverse have the same order, you will usually have to
deal with the finite case and infinite case separately. That is, *|a| = n* is one
argument and *|a|* is infinity is a different case. This is usually
true as well
when dealing with the order of a group. The cases of a finite group and an
infinite group may require different arguments.

15. A number of exercises ask for a yes or no answer. Provide a reason for your answer. For instance, for Exercise 23 of Chapter 2 specify
a group and elements *a* and *b* such that *(ab) ^{2}* is not equal to

16. In general, you cannot take roots (square roots, cube roots, etc.) in groups. Only integer powers of group elements are permissible.

17. Whenever you are asked to prove a subset of a group is a subgroup, use one of the subgroups tests. If you know the subset is finite use the Finite Subgroup Test.

18. When you are asked to prove that some integer *a* is divisible by an integer *b* write *a = bq + r* where *r*
greater than or equal to 0 and less than *b* and use what you are given to show that * r = 0*.
See Exercise 59 in Chapter 3 [Exercise 61 8th ed.] for an example.

19. When an exercise says prove something is true for an integer do not assume
the integer is positive. In general, the cases that an integer is positive and
an integer is negative require slightly different arguments. Usually, you can
use the positive integer case to prove the negative integer case by using the
Law of Exponents. To illustrate the technique consider Exercise 27 in Chapter
2. To prove * (a ^{-1}ba)^{n} = a^{-1}b^{n}a
*for all

(a

20. When dealing with an abstract group (that is, one in which the elements and
operation are not specified) use *e* to denote the identity and use
multiplication
as the operation (that is, *ab*). If you are told the operation is addition use * a + b *, *-a* for the inverse of *a*, and 0 for
the group identity.

21. The negation "for all" is "there exist some." For example, in an
Abelian group *ab = ba* for all *a* and *b*. So, in a
non-Abelian group there
exist SOME elements *a* and *b* such that *ab* is not *ba*.
To remember this think of a common statement such as "The team won every game."
The negation is "There exist some game the team did not win."

22. When ask to prove two groups are not isomorphic students often show
that some specific mapping does not satisfy the definition of isomorphism.
This merely proves that specific mapping is not an isomorphism. It does not preclude that some other mapping may be an isomorphism.
Instead, one must show
that NO mapping satisfies the
definition. This can be done by assuming there is some generic isomorphism
and using only properties of isomorphisms derive a contradiction.
Example 5 of Chapter 6 illustrates how this can be done. Notice that
no specific mapping was assumed. Usually the easiest way to prove that two
groups are not isomorphic is to show that they do not share some group
property. For example, the group of nonzero complex numbers under
multiplication has an element of order 4 (the square root of -1) but the
group of nonzero real numbers do not have an element of order 4. As
another example, we see that *S _{4}* is not isomorphic to