Lecture 1

Free Product of Groups. Let A, B be groups, such that A ∩ B = ∅.

Definition 1. The free product of groups A, B is a group A ∗ B given by the following:

Elements of A ∗ B are equivalence classes of finite sequences in A ⊔ B.

Equivalence relation is generated by two “elementary” equivalences.

“Removing trivial element”: V 1_{A}W ∼ V 1_{B}W ∼ V W.

“Combining together”: V g_{1}g_{2}W ∼ V (g_{1}g_{2})W, provided g_{1} and g_{2} both in A or both in B.

Multiplication is concatenating representatives:

[g_{1}g_{2} ⋯ g_{m}][h_{1}h_{2} ⋯ h_{n}] = [g_{1}g_{2} ⋯ g_{m}h_{1}h_{2} ⋯ h_{n}].

Examples of elements of A ∗ B: g_{1}g_{2}g_{3}, 1 (empty sequence), g_{1}.

Lemma 1. Multiplication is well-defined.

Proof. An exercise. □

One can easily see that A ∗ B is a group indeed:

multiplication is obviously associative;

identity is [1];

inverse element is given by [g_{1}g_{2} ⋯ g_{m}]^{−1} = [g_{m}^{−1} ⋯ g_{2}^{−1}g_{1}^{−1}].

Lemma 2. Both A and B embed in A ∗ B.

Proof. An exercise. □

Let A be presented by ⟨a_{1}, … |R_{1}, …⟩, and B be presented by ⟨b_{1}, … |S_{1}, …⟩. Then A ∗ B is presented by ⟨a_{1}, …, b_{1}, … |R_{1}, …, S_{1}, …⟩.

Universal Property. Free product A ∗ B satisfies the following universal property: for any group G and homomorphisms A → G, B → G there exists a unique homomorphism A ∗ B → G such that the diagram commutes.

\xymatrix{A \ar@{^{(}->}[d] \ar[dr] & B \ar@{^{(}->}[dl] \ar[d] \\ A ∗ B \ar@{-->}[r] & G}

In other words, A ∗ B is a universal sender in the category “pairs of homos A → G, B → G”.

Therefore, we effectively can define a free product in four ways: constructively, in terms of presentations, by a universal property, and in terms of theory of categories.

Notable examples of free products: F_{2} ≅ Z ∗ Z, or ⟨a, b|−⟩ ≅ ⟨a|−⟩ ∗ ⟨b|−⟩; Z_{2} ∗ Z_{3}.

Amalgamated Product of Groups. Let A, B, C be groups such that A∩B = ∅. Let Φ_{A} : C → A, Φ_{B} : C → B be isomorphic embeddings.

Definition 2. The amalgamated product of groups A, B along C is a group A ∗_{C} B containing natural copies of A, B, characterized by the following property: for each commutative diagram of homomorphisms (see left) there exists a unique homomorphism A \begin{matrix}∗ \\ C \end{matrix} B → G such that diagram commutes (see right).

\xymatrix{ & A \ar[dr] &  &  & A \ar[dr] \ar[drr] \\ C \ar@{^{(}->}[ur] \ar@{^{(}->}[dr] &  & G & C \ar@{^{(}->}[ur] \ar@{^{(}->}[dr] &  & A ∗_{C} B \ar@{-->}[r] & G \\  & B \ar[ur] &  &  & B \ar[ur] \ar[urr]}

In other words, an amalgamated product is something like a free product of groups which share a common subgroup.

Let A be presented by ⟨a_{1}, … |R_{1}, …⟩, B be presented by ⟨b_{1}, … |S_{1}, …⟩, and C be generated by ⟨c_{1}, …⟩. Then A \begin{matrix}∗ \\ C \end{matrix} B is presented by

⟨a_{1}, …, b_{1}, … |R_{1}, …, S_{1}, …, Φ_{A}(c_{i}) = Φ_{B}(c_{i}) ∀i⟩.

Exercise. Prove that.

Alternate Construction of Amalgamated Product. One can provide an alternate construction of A \begin{matrix}∗ \\ C \end{matrix} B.

Definition 3. Let A, B be groups such that A ∩ B = ∅. Let C_{A} ⊆ A, Let C_{B} ⊆ B, and C_{A} ≅ C_{B} ≅ C. Then A \begin{matrix}∗ \\ C \end{matrix} B = A \begin{matrix}∗ \\ C_{A}=C_{B} \end{matrix} B is given by the following:

Elements of A ∗ B are equivalence classes of finite sequences in A ⊔ B.

Equivalence relation is generated by two “elementary” equivalences.

“Combining together”: V g_{1}g_{2}W ∼ V (g_{1}g_{2})W, provided g_{1} and g_{2} both in A or both in B.

“Sharing a subgroup”: V Φ_{A}(c)W ∼ V Φ_{B}(c)W.

Multiplication is concatenating representatives:

[g_{1}g_{2} ⋯ g_{m}][h_{1}h_{2} ⋯ h_{n}] = [g_{1}g_{2} ⋯ g_{m}h_{1}h_{2} ⋯ h_{n}].

An example: g_{1}g_{2} = (g_{1}1_{B})g_{2} = g_{1}1_{B}g_{2} = g_{1}1_{A}g_{2}.

Lemma 3. Multiplication is well-defined.

Proof. An exercise. □

One can easily see that A \begin{matrix}∗ \\ C \end{matrix} B is really a group:

multiplication is obviously associative;

identity is [1_{B}] = [1_{A}];

inverse element is given by [g_{1}g_{2} ⋯ g_{m}]^{−1} = [g_{m}^{−1} ⋯ g_{2}^{−1}g_{1}^{−1}]. Note that the empty sequence is not considered an element.

Normal Form Theorem. A normal form is a particular type of sequence (representing some element) g_{0}g_{1}g_{2} ⋯ g_{k}:

• g_{0} is either in A or B;

• (i ≥ 0) g_{i}, g_{i+1} are not both in A or both in B;
• (i ≥ 1) g_{i} is not in Φ_{A}(C) or Φ_{B}(C).

The number k is called length of the sequence.

Theorem 1. 1) Two normal forms representing the same element have the same length.

2) 1_{A} and 1_{B} are the only normal forms for 1.

3) If g_{0} and h_{0} are normal forms for some element, then either g_{0} = h_{0}, or g_{0} = Φ_{A}(c) and h_{0} = Φ_{B}(c) for some c ∈ C (or vice versa).

Remark. The theorem does not seem true. More exactly, the NFT is fine, but something is wrong with the definition of the normal form. For example, any element a ∈ A − Φ_{A}(C) has the normal form of the length 0: a, and one of the length 1: Φ_{B}(c)·(Φ_{A}(c^{−1})a. Maybe it should be said explicitly that the formal form starts with an element either of A − Φ_{A}(C) or of B − Φ_{B}(C), then the numeration of g’s starts with 1 instead of 0.

Corollary 1. A, B imbed in A \begin{matrix}∗ \\ C \end{matrix} B. Moreover, A ∩ B ≅ C.