MATH 430 PART 2: GROUPS AND SUBGROUPS Last class, we encountered the structure D 3 where the set was motions which preserve an equilateral triangle and the operation was function composition. We determined that this structure had certain collection of handy properties that we like. This type of structure will be our new object of study, and our main focus for this semester. 1. Groups Definition 1. A group is a set G with a binary operation, such that G is closed under, is associative on G, has an identity element on G, and all elements of G have an inverse under. Example 1. Each of the following is a group: D 3 (the symmetries of an equilateral triangle) Z, Q, R, or C under addition M m n (R) (m n matrices with real entries) under matrix addition Exercise 1. Each of the following is NOT a group. For each, determine all the properties which do not hold. Z under subtraction Z under multiplication Q, R, or C under multiplication M 2 (R) (2 2 matrices with real entries) under matrix multiplication 1
2 MATH 430 PART 2: GROUPS AND SUBGROUPS Exercise 2. The set {1, 1, i, i} C under multiplication certainly satisfies associative and identity. Prove that the other two properties also hold. (Hint: Write out the multiplication table!) Example 2. The set {0, 1, 2, 3, 4, 5} under addition mod 6 is a group: Closed: Adding two integers mod 6 produces an element of this set. (You could write out the table for the operation if you wanted to.) Associative: Addition mod 6 is a known associative operation. Identity: 0 is the identity for addition mod 6. Inverses: In this set, 0 and 3 are their own inverses, 1 and 5 are inverses, and 2 and 4 are inverses. (You could point these out in the table for the operation if you wanted to.) Important Remark: We name this group Z/6Z but your book calls it Z 6. In fact there is a whole family of groups Z/nZ for positive integers n where the set is {0, 1, 2,... n 1} and the operation is addition mod n. Exercise 3. Let GL n (R) be the set of n n real matrices with nonzero determinant under multiplication. Prove that this is a group.
MATH 430 PART 2: GROUPS AND SUBGROUPS 3 Exercise 4. Determine if Q + under the operation a b = ab 2 properties fail. is a group. If so, prove it; if not, identify which Example 3. The set {0, 1, 2, 3, 4, 5, 6, 7} under multiplication mod 8 is NOT a group. The elements 0, 2, 4, and 6 have no inverse under this operation (make the table for the operation to see this). However, the set {1, 3, 5, 7} IS a group under multiplication mod 8. (You ll see this in homework by constructing the multiplication table.) Example 4. The group of symmetries on any regular n-gon is a group. We call it D n. (So D 17 is the group of actions on a regular 17-gon that preserve the polygon, much like what we did for an equilateral triangle in class.) Here are our first adjectives that can be applied to groups. Definition 2. If is commutative on a group G, then we say G is abelian. Example 5. Z, Q, R, and C under addition, Z/nZ, and {1, 1, i, i} under multiplication are all examples of abelian groups. GL n (R) and D 3 are examples of nonabelian groups. Definition 3. The order of a group is the number of elements in that group. (A group with infinitely many elements is called a group of infinite order.) We denote the order of G by G. Example 6. GL 2 (R) is of infinite order and Z/6Z = 6. Exercise 5. With a partner or two, do these. (a) Prove that a group G with identity e such that a 2 = e for all a G is abelian. (Hint: Consider (ab) 2.) (b) Let G be a group of finite order. Show that for any a G there exists an n Z + such that a n = e. (Hint: Consider the list e, a, a 2,..., a m where m is the number of elements in G, and use the cancellation laws.)
4 MATH 430 PART 2: GROUPS AND SUBGROUPS 2. Properties of groups Because we insist on associativity, identity, and inverses for a structure to be a group, there are certain properties that all groups share no matter what kind of group they are. Theorem 1. If G is a group under then for any a, b, c G, if a b = a c then b = c. Similarly, if b a = c a then b = c. (We call these properties left and right cancellation, respectively.) Proof. (Prove the left cancellation property using inverses.) Exercise 6. The cancellation properties have a very important impact on how the operation acts on G. a b c Use it to complete this group table for G = {a, b, c} (assume a b c): a c b b a b c c a Theorem 2. If G is a group under and a, b G, then the linear equations a x = b and y a = b have unique solutions x and y in G. Proof. (First show there is a solution to a x = b, then show by contradiction that there can t be more.)
MATH 430 PART 2: GROUPS AND SUBGROUPS 5 Theorem 3. Let G be a group. The identity of G is unique, and for all a G, a 1 is unique. Proof. The identity of G is unique because G is a binary structure and we proved in Theorem 2 on page 4 of Part 1 that binary structures have at most one identity element. To show inverses are unique, we suppose that b and c are both inverses of a in G. Then a b = e and a c = e by definition of inverse. But a b = a c b = c by left cancellation, so the inverse of a is unique. Important Remark on Notation: Writing the operation is getting old and is a waste of effort. From here on out, any generic/abstract operation (like ) will be written as juxtaposition. In other words, now we re going to write ab instead of a b. (We ll still write + for addition (regular or mod n) though.) Exercise 7. If G is a group and a, b G, certainly ab is also an element of G. What is the inverse of ab? (In other words, find (ab) 1 in G.) Exercise 8. With a partner or two, try these problems from section 4: #32, 34, 36, 37. I ll choose groups at random to share solutions, so make sure you have nicely written proofs! 3. Small finite groups Example 7. The group of order one contains only the identity, {e}. This is the unique group of order one. Exercise 9. Describe a group of order two. What are its elements? Can you write a group table? How many distinct (non-isomorphic) groups of order two are there?
6 MATH 430 PART 2: GROUPS AND SUBGROUPS Exercise 10. Find all distinct groups of order three, giving a Cayley table for each one. Exercise 11. Name two non-isomorphic groups of order 6. How do you know they re not isomorphic? 4. Subgroups Definition 4. If H is a subset of a group G and H is also a group under the same operation as G, then we call H a subgroup of G and we write H G. If H G we can write H < G and we call H a proper subgroup of G. Example 8. R and C are both groups under normal addition, so (R, +) < (C, +). Z/5Z is not a subgroup of Z/7Z even though {0, 1, 2, 3, 4} {0, 1, 2, 3, 4, 5, 6} because the operations + mod 5 and + mod 7 are not the same. Example 9. The group {e} containing only the identity is a proper subgroup of every group. We call this group the trivial group (or the trivial subgroup). All other subgroups are nontrivial subgroups. Example 10. The Klein 4 group, denoted V is a group of order four with the following Cayley table: e a b c e e a b c a a e c b b b c e a c c b a e The group V has nontrivial subgroups {e, a}, {e, b}, {e, c}.
MATH 430 PART 2: GROUPS AND SUBGROUPS 7 Exercise 12. Find all subgroups of Z/4Z. A subgroup diagram is a graph which shows how all subgroups of G are related with respect to containment. Here s the subgroup diagram for V, the Klein 4 group. Exercise 13. Draw the subgroup diagram for Z/4Z. V {e, a} {e, b} {e, c} {e} Note: Based on their subgroup diagrams, we can tell that the group V is not isomorphic to Z/4Z, so there are two non-isomorphic groups of order 4! Theorem 4. A subset H of a group G is a subgroup if and only if (1) H is closed under the same operation as G; (2) H contains the identity of G (e H); (3) for all a H, a 1 H also. Proof. This is three of the four conditions for H to be a group anyway. The operation on G is associative by hypothesis; since H G, the operation is still associative on H. There s a shorter criterion for showing a subset is a subgroup, which can be nice in some cases. (You may use either, and I will never specify which I want you to use.) Theorem 5 (2-step subgroup test). A subset H of a group G is a subgroup of G if and only if H is nonempty and ab 1 H for all a, b H.
8 MATH 430 PART 2: GROUPS AND SUBGROUPS Exercise 14. Show that nz = {nm m Z} is a subgroup of Z for all n Z. Question: Is 4Z a subgroup of 2Z, or is 2Z a subgroup of 4Z, or neither? In general, when is nz < mz? Exercise 15. Prove that SL n (R) is a subgroup of GL n (R). Exercise 16. With a partner or two, do these. I ll choose groups at random to share solutions, so make sure you have nicely written proofs! (a) Let F be the group of all real-valued functions with domain R under addition. Prove that the set of functions H = {f F f(1) = 0} is a subgroup of F. (b) Let G be a group and let a be a fixed element of G. Show that C(a) = {x G xa = ax} is a subgroup of G. (C(a) is called the centralizer of a.) (c) Suppose that H and K are both subgroups of an abelian group G. Prove that S = {hk h H, k K} is a subgroup of G. (Note: In this problem, you need to also explain why S is even a subset of G as part of your proof that it is a subgroup!)