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Home , Examples of groups

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 ).

GLn(R) = set of all n × n invertible matrices with R-entries = {M ∈ Matn(R) | det(M) 6= 0} n n = set of bijective linear transformations T : R → R

The GLn(R) under multiplication is a group. WHY?

• A, B ∈ GLn(R) =⇒ AB ∈ 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 , which is not invertible! Easiest proof: use fact that det(AB) = det(A) det(B).

• A, B, C ∈ 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 ∈ GLn(R).

−1 • A ∈ GLn(R) =⇒ A exists and is unique.

Example 1.2 ().

SLn(R) = set of all n × n matrices with R-entries and 1 = {M ∈ Matn(R) | det(M) = 1}

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 ): det : GLn(R) −→ R −{0}. We use the symbol R −{0} to mean the multiplicative group of nonzero real numbers. Other × equivalent notations for the set of nonzero real numbers are R \{0} or R .

ELFS: Explain why ker(det) = SLn(R).

Example 1.4 (Integers under addition (Z, +)). The set Z = {..., −2, −1, 0, 1, 2,...} under addition is a group. WHY?

• a, b ∈ Z =⇒ a + b ∈ Z

• a, b, c ∈ Z =⇒ a + (b + c) = (a + b) + c

• There exists a unique identity element 0 ∈ Z such that a + 0 = 0 + a for all a ∈ Z.

• a ∈ 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 ∈ Z. Is there an integer m ∈ 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 ∈ V =⇒ ~v + ~w ∈ V

• ~u,~v, ~w ∈ V =⇒ ~u + (~v + ~w) = (~u + ~v) + ~w

• There exists a unique identity vector ~0 ∈ V such that ~v + ~0 = ~0 + ~v for all ~v ∈ V .

• ~v ∈ V =⇒ −~v exists and is unique.

2 Example 1.7 (The Sym(T ) of a set T ). Let T be a set. Define a map G as follows:

G = {g : T → T | g is a bijection} = 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 ∈ G =⇒ g ◦ h ∈ G

• g, h, k ∈ G =⇒ g ◦ (h ◦ k) = (g ◦ h) ◦ k

• There exists a unique identity map i ∈ G given by i : T → T such that i(t) = t for all t ∈ T . Moreover g ◦ i = i ◦ g for all g ∈ G.

• g ∈ 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 = {1, 2, . . . , n}, then Sym(T ) is called the 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 n is a of the symmetric group Sn.

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