Math 425 - 1st day of class Notes by Prof aBa typed via TEX Reminder to students: Feel free to raise your hand and ask for a minute of silence if you need to catch up on taking notes or digest something. NOTE: I will use the term ELFS to mean Exercise Left For Students. 1 Examples of groups Example 1.1 (General Linear Group). GLn(R) = set of all n × n invertible matrices with R-entries = fM 2 Matn(R) j det(M) 6= 0g n n = set of bijective linear transformations T : R ! R The general linear group GLn(R) under multiplication is a group. WHY? • A; B 2 GLn(R) =) AB 2 GLn(R) ELFS: How do we know that the product of two invertible matrices is still 1 0 −1 0 invertible? For sums, it is not true { for example, the sum [ 0 1 ] + 0 −1 equals the zero matrix, which is not invertible! Easiest proof: use fact that det(AB) = det(A) det(B). • A; B; C 2 GLn(R) =) A(BC) = (AB)C What is this property called? • There exists a unique identity element In such that AIn = InA for all A 2 GLn(R). −1 • A 2 GLn(R) =) A exists and is unique. Example 1.2 (Special Linear Group). SLn(R) = set of all n × n matrices with R-entries and determinant 1 = fM 2 Matn(R) j det(M) = 1g n n It is also the set of bijective linear transformations T : R ! R that preserve volume and orientation. NOTE: I did not talk about this final characterization. The special linear group SLn(R) under multiplication is a group. WHY? ELFS: Show that SLn(R) satisfies the same four bullet points that GLn(R) satisfied in the previous example. 1 Remark 1.3. The group SLn(R) is the kernel of the following determinant map (recall your linear algebra): det : GLn(R) −! R −{0g: We use the symbol R −{0g to mean the multiplicative group of nonzero real numbers. Other × equivalent notations for the set of nonzero real numbers are R nf0g or R . ELFS: Explain why ker(det) = SLn(R). Example 1.4 (Integers under addition (Z; +)). The set Z = f:::; −2; −1; 0; 1; 2;:::g under addition is a group. WHY? • a; b 2 Z =) a + b 2 Z • a; b; c 2 Z =) a + (b + c) = (a + b) + c • There exists a unique identity element 0 2 Z such that a + 0 = 0 + a for all a 2 Z. • a 2 Z =) −a exists and is unique. Question 1.5. Why is (Z; ×) not a group? It seems like it would be group because it satisfies the first three of the four bullet points above if we replace the operation + with the operation ×, and replace the number 0 with the value 1. But it fails to satisfy the fourth bullet point. Consider the element 7 2 Z. Is there an integer m 2 Z such that 7 × m = 1? Example 1.6 (Vector space V over a field F ). Recall from linear algebra the definition of a vector space V over a field F . We claim that if we ignore the scalar multiplication property, V becomes a vector space under the operation vector addition. WHY? • ~v; ~w 2 V =) ~v + ~w 2 V • ~u;~v; ~w 2 V =) ~u + (~v + ~w) = (~u + ~v) + ~w • There exists a unique identity vector ~0 2 V such that ~v + ~0 = ~0 + ~v for all ~v 2 V . • ~v 2 V =) −~v exists and is unique. 2 Example 1.7 (The symmetry group Sym(T ) of a set T ). Let T be a set. Define a map G as follows: G = fg : T ! T j g is a bijectiong = Sym(T ) Question: What are the two conditions that a bijective map satisfies? We claim that G is a group under composition of maps. WHY? • g; h 2 G =) g ◦ h 2 G • g; h; k 2 G =) g ◦ (h ◦ k) = (g ◦ h) ◦ k • There exists a unique identity map i 2 G given by i : T ! T such that i(t) = t for all t 2 T . Moreover g ◦ i = i ◦ g for all g 2 G. • g 2 G =) g−1 exists in G and is unique. ELFS: Explain why G satisfies the four bullet points above. HINT: use the fact that bijective maps possess the two conditions in the question above. 2 Cayley's Theorem If T = f1; 2; : : : ; ng, then Sym(T ) is called the symmetric group and is denoted Sn. That symbol Sn in TEX is the font math fraktur, and the symbol is a fraktur-S with a subscript n. Some write Sn, but more commonly it is written as Sn. Theorem 2.1. Every finite group of order n is a subgroup of the symmetric group Sn. 3.