Review of normal subgroups, cosets, quotients, and generators for subgroups and normal subgroups
Definition5.4.1.
Let \(A\) be a set, let \(A' :=\{a-1 | a \in A\}\) be a set that bijects to \(A\text{,}\) let \((A \cup A')*\) be the set of all words written with the letters of \(A \cup A'\) (including the empty word, denoted \(1\)), and let \sim be the smallest equivalence relation on (A \cup A')* such that xaa-1y \sim xy \sim xa-1ay for all a \in A and x,y \in (A \cup A')*. The free group on A, denoted F(A), is the quotient set (A \cup A')*/\sim with the group operation [v][w] := [vw] where vw is the concatenation of the words v and w. In the case that |A| = n, this group is also denoted Fn.
Definition5.4.2.
Let \(A\) be a set and let \(R\) be a subset of \(F(A)\text{.}\) The normal subgroup generated by \(R\) is \(\langle R \rangle^N :=\{u1r1e1u1-1 ··· ukrkekuk-1 | k \geq 0, and ri \in R, ei \in\{1,-1\}, and ui \in F(A) for each 1 \leq i \leq k\}\text{.}\)
Definition5.4.3.
Let \(A\) be a set and let \(R\) be a subset of \(F(A)\text{.}\) The group presented by the presentation \(\langle A | R \rangle\) is the quotient group \(F(A)/\langle R \rangle^N\text{.}\) The set \(A\) is the set of generators, the set \(R\) is the set of relators, and the set of equations \(\{r = 1 | r \in R\}\) is the set of relations of the presentation.
Definition5.4.4.
For a set \(A, R \sse F(A)\text{,}\) and words \(v,w \in (A \cup A')*\text{,}\) the equation \(v = w\) means that \(v\) and \(w\) are the same word, \(v =F(A) w\) means that \([v] = [w]\) in the group \(F(A)\text{,}\) and \(v =G w\) means that \([v]\langle R \rangle^N = [w]\langle R \rangle^N\) in the group \(G := \langle A | R \rangle\text{.}\)
Remark5.4.5.
The group \(\langle A | R \rangle\) is the largest group generated by \(A\) satisfying \(r =G 1 for all r \in R\text{.}\)
Definition5.4.6.
A group \(G\) is finitely generated if \(G\) is a quotient of \(F(A)\) for some finite set \(A\text{,}\) and \(G\) is finitely presented if \(G = \langle A | R \rangle\) for some finite sets \(A\) and \(R\text{.}\)
Example5.4.7.
Examples
Lemma5.4.8.
If \(G\) is a group, then \(G\) has a presentation; moreover, \(G\) is presented by \(G = \langle G | ab = (ab)\) for all \(a,b \in G \rangle\text{.}\)
Theorem5.4.9.HBTP = "Homomorphism Building Theorem for presentations".
Let \(G = \langle A | R \rangle\text{,}\) let \(H\) be a group, and let \(f:A \to H\) be a function satisfying the property that for all words \(b1e1 ··· bmem \in R\) (with each \(bi \in A\) and \(ei \in\{1,-1\}), f(b1)e1 ··· f(bm)em =H 1\text{.}\) Then there is a unique group homomorphism \(h:G \to H satisfying h(a) = f(a)\) for all \(a \in A\text{.}\)
Definition5.4.10.
Let \(A\) be a set, let \(R \sse F(A)\text{,}\) let \(b\) be a letter not in \(A\text{,}\) let \(w \in F(A)\text{,}\) and let \(r \in \langle R \rangle^N\text{.}\) The operations \(\langle A | R \rangle ↔ \langle A \cup\{b\} | R \cup\{b = w\} \rangle\) and \(\langle A | R \rangle ↔ \langle A | R \cup\{r\} \rangle\) are Tietze transformations.
Theorem5.4.11.
If \(\langle A | R \rangle \cong \langle B | S \rangle\text{,}\) then there is a finite sequence of Tietze transformations from \(\langle A | R \rangle\) to \(\langle B | S \rangle\text{.}\)
Subsection5.4.2Building new groups from old -- or decomposing groups
SubsubsectionAbelianization
Definition5.4.12.
The abelianization of a group \(G\) is the quotient group \(Gab := G/[G,G]\) where \([G,G]\) is the commutator subgroup \([G,G] :=\{aba-1b-1 | a,b \in G\}\text{.}\) The element \(aba-1b-1\) of \(G\) is denoted \([a,b]\) and called the commutator of \(a\) with \(b\text{.}\)
Remark5.4.13.
There are \(3\) views of \(Gab\text{:}\) Using presentations (if \(G = \langle A | R \rangle\text{,}\) then \(Gab = \langle A | R \cup\{aba-1b-1 | a,b \in A\} \rangle\)), as the quotient of \(G\) by its commutator subgroup (as in Def 5.115), and as the largest abelian group that is a quotient of \(G\text{.}\)
Proposition5.4.14.
If \(G\) and \(H\) are groups and \(Gab ≇ Hab\text{,}\) then \(G ≇ H\text{.}\)
Example5.4.15.
Examples
SubsubsectionDirect products
Definition5.4.16.
Let \(g_\alpha\) be a group, and write \(g_\alpha = \langle A_\alpha | R\alpha \rangle\text{,}\) for each \(\alpha\text{.}\) The direct sum of the \(g_\alpha\) is the \(group \oplus g_\alpha = \langle \cup \alpha A_\alpha | \cup \alpha R\alpha \cup\{ab = ba | a \in g_\alpha, b \in \alpha, \alpha \ne \beta \} \rangle\text{.}\)
Remark5.4.17.
There are \(3\) views of \(G \times H\text{:}\) Using presentations (if \(G = \langle A | R \rangle\) and \(H = \langle B | S \rangle\text{,}\) then \(G \times H = \langle A \cup B | R \cup S \cup\{ab = ba | a \in A, b \in B\}\rangle\)), as a Cartesian product set with componentwise multiplication, and as the largest group generated by \(G\) and \(H\) such that the subgroups \(G\) and \(H\) commute.
Example5.4.18.
Examples
SubsubsectionFree Products
Definition5.4.19.
Let \(g_\alpha\) be a group, and write \(g_\alpha = \langle A_\alpha | R\alpha \rangle\text{,}\) for each \(\alpha\text{.}\) The free product of the \(g_\alpha\) is the group \(∗\alpha g_\alpha = \langle \cup \alpha A_\alpha | \cup \alpha R\alpha \rangle\text{.}\)
Definition5.4.20.
Let \(g_\alpha\) be a group for all \(\alpha\text{.}\) A reduced sequence for the collection of groups \(g_\alpha\) is a sequence of group elements (or word) of the form \(g1 ··· gk\) such that \(k \geq 0\text{,}\) for each \(i \in\{1,\dots,k\}\) there is an index \(\alpha i\) such that \(gi \in g_\alpha i\setminus\{1g_\alpha i\}\text{,}\) and for each \(i \in\{1,\dots,k-1\}\text{,}\)\(\alpha i \ne \alpha i+1\text{.}\) In the case of two groups \(G\) and \(H\text{,}\) a reduced sequence for \(G,H\) is a word of one of the forms \(g1h1 ··· gkhk, g1h1 ··· hk-1gk, h1g2 ··· gkhk\text{,}\) or \(h1g2 ··· hk-1gk\text{,}\) such that \(k \geq 0\text{,}\) and each \(gi \in G\setminus\{1G\}\) and \(hi \in H\setminus\{1H\}\text{.}\)
Lemma5.4.21.
If \(G\) and \(H\) are groups, then \(G ∗ H\) is isomorphic to the set of reduced sequences for \(G,H\) with group operation given by concatenation and reduction (in the groups \(G\) and \(H\)) to a reduced sequence.
Remark5.4.22.
There are \(3\) views of \(G ∗ H\text{:}\) Using presentations (if \(G = \langle A | R \rangle\) and \(H = \langle B | S \rangle\text{,}\) then \(G ∗ H\langle = \langle A \cup B | R \cup S \rangle\)), as the set of reduced sequences for \(G,H\) (as in Lemma 5.105), and as the largest group generated by \(G\) and \(H\text{.}\)
Example5.4.23.
Examples
Theorem5.4.24.HBTFP = "Homomorphism Building Theorem for free products".
Let \(g_\alpha\) be a group for each \(\alpha\text{,}\) let \(J\) be a group, and let \(f_\alpha :g_\alpha \to J\) be a homomorphism for each \(\alpha\text{.}\) Then there is a unique group homomorphism \(h:∗\alpha g_\alpha \to J satisfying h(g) = f(g)\) for all \(g \in g_\alpha\text{,}\) for all \(\alpha\text{.}\)
Example5.4.25.
Examples
SubsubsectionFree products with amalgamation
Definition5.4.26.
Let G, H, and K be groups, and let r:K \to G and s:K \to H be group homomorphisms. The associated free product with amalgamation, or amalgamated product, is the quotient group G ∗K H := (G ∗ H) / \langle{r(k)s(k)-1 | k \in K\}\rangle^N.
Remark5.4.27.
There are 3 views of G ∗K H: Using presentations (if G = \langle A | R \rangle, H = \langle B | S \rangle, and K = \langle C \rangle, then G ∗K H = \langle A \cup B | R \cup S \cup\{r(k)s(k)-1 | k \in K\}\rangle), using cosets represented by reduced sequences, and as the largest group generated by G and H glued along the common subgroups r(K) in G and s(K) in H.
Theorem5.4.28.HBTAP = "Homomorphism Building Theorem for amalgamated products".
Let G, H, K, and J be groups, and r:K \to G, s:K \to H, a:G \to J and b:H \to J be homomorphisms satisfying the property that a \circ r = b \circ s. Then there is a unique group homomorphism c:G ∗K H \to J satisfying c(g) = a(g) for all g \in G and c(h) = b(h) for all h \in H.
