Homologism of groups: Difference between revisions

Latest revision as of 04:18, 27 May 2013

Definition

Definition in terms of a defining set of words

Consider a subvariety $V$ of the variety of groups. Denote by $W$ a set of words that generate the variety $V$ (i.e., a group is in $V$ iff all words from $W$ are trivial for all tuples of elements from the group).

Consider any group $G$ (not necessarily in $V$ ). Denote by $V^{*} (G)$ the marginal subgroup of $G$ with respect to the variety $V$ and denote by $V (G)$ the verbal subgroup of $G$ with respect to $V$ .

For every word $w \in W$ , let $n_{w}$ be the number of distinct letters used in the word. $w$ defines a $n_{w}$ -ary set map:

$β_{w, G} : G^{n_{w}} \to G$

By the definitions of marginal and verbal subgroup, the map descends to a set map:

$γ_{w, G} : (G / V^{*} (G))^{n_{w}} \to V (G)$

A homologism of groups $G_{1}$ and $G_{2}$ with respect to $V$ is a pair $(ζ, φ)$ where $ζ : G_{1} / V^{*} (G_{1}) \to G_{2} / V^{*} (G_{2})$ , $φ : V (G_{1}) \to V (G_{2})$ are homomorphisms, and for every $w \in W$ , we have:

$γ_{w} (ζ (x_{1}), ζ (x_{2}), \dots, ζ (x_{n_{w}}) = φ (γ_{w} (x_{1}, x_{2}, \dots, x_{n_{w}})) \forall (x_{1}, x_{2}, \dots, x_{n}) \in (G_{1} / V^{*} (G_{1}))^{n_{w}}$

Note that the choice of $W$ does not matter for this definition, all that matters is that $W$ generate the variety $V$ .

In other words, for every $w \in W$ , the following diagram commutes:

$\begin{matrix} (G_{1} / V^{*} (G_{1}))^{n_{w}} & (G_{2} / V^{*} (G_{2}))^{n_{w}} \\ ↓^{γ_{w, G_{1}}} & ↓^{γ_{w, G_{2}}} \\ V (G_{1}) & V (G_{2}) \end{matrix}$

Note that the choice of the defining set of words does not matter, i.e., if $W_{1}$ and $W_{2}$ are different sets of words that generate the same variety $V$ , the condition of being a homologism with respect to $W_{1}$ coincides with the condition of being a homologism with respect to $W_{2}$ .

Related notions

Term	Meaning
category of groups with homologisms	this is a category whose objects are group and where the morphisms are homologisms of groups.
isologism of groups	this is a homologism where both the component homomorphisms are isomorphisms.

Particular cases

Variety	Generating word or set of words	Corresponding notion of homologism
variety of abelian groups	$[x_{1}, x_{2}]$ -- the commutator word	homoclinism of groups
variety of groups of nilpotency class at most $n$	$[[\dots [[x_{1}, x_{2}], x_{3}], \dots, x_{c}], x_{c + 1}]$ -- the left-normed iterated commutator word	n-homoclinism of groups

@@ Line 9: / Line 9: @@
 For every word <math>w \in W</math>, let <math>n_w</math> be the number of distinct letters used in the word. <math>w</math> defines a <math>n_w</math>-ary set map:
-<math>\beta_w: G^{n_w} \to G</math>
+<math>\beta_{w,G}: G^{n_w} \to G</math>
 By the definitions of marginal and verbal subgroup, the map descends to a set map:
-<math>\gamma_w: (G/V^*(G))^{n_w} \to V(G)</math>
+<math>\gamma_{w,G}: (G/V^*(G))^{n_w} \to V(G)</math>
-A '''homologism''' of groups <math>G</math> and <math>H</math> with respect to <math>\mathcal{V}</math> is a pair <math>(\zeta,\phi)</math> where <math>\zeta</math> is a homomorphism between <math>G/V^*(G)</math> and <math>H/V^*(H)</math>, <math>\phi</math> is a homomorphism between <math>V(G)</math> and <math>V(H)</math>, and for every <math>w \in W</math>, we have:
+A '''homologism''' of groups <math>G_1</math> and <math>G_2</math> with respect to <math>\mathcal{V}</math> is a pair <math>(\zeta,\varphi)</math> where <math>\zeta: G_1/V^*(G_1) \to G_2/V^*(G_2)</math>, <math>\varphi: V(G_1) \to V(G_2)</math> are homomorphisms, and for every <math>w \in W</math>, we have:
-<math>\gamma_w(\zeta(x_1), \zeta(x_2), \dots, \zeta(x_{n_w}) = \phi(\gamma_w(x_1,x_2,\dots, x_{n_w})) \ \forall \ (x_1,x_2,\dots,x_n) \in (G/V^*(G))^{n_w}</math>
+<math>\gamma_w(\zeta(x_1), \zeta(x_2), \dots, \zeta(x_{n_w}) = \varphi(\gamma_w(x_1,x_2,\dots, x_{n_w})) \ \forall \ (x_1,x_2,\dots,x_n) \in (G_1/V^*(G_1))^{n_w}</math>
 Note that the choice of <math>W</math> does not matter for this definition, all that matters is that <math>W</math> generate the variety <math>\mathcal{V}</math>.
+In other words, for every <math>w \in W</math>, the following diagram commutes:
+<math>\begin{array}{ccc}
+  (G_1/V^*(G_1))^{n_w} & \stackrel{\zeta \times \zeta}{\to} & (G_2/V^*(G_2))^{n_w} \\
+  \downarrow^{\gamma_{w,G_1}}  & & \downarrow^{\gamma_{w,G_2}}\\
+  V(G_1) & \stackrel{\varphi}{\to} & V(G_2)\\
+\end{array}</math>
+Note that the choice of the defining set of words does not matter, i.e., if <math>W_1</math> and <math>W_2</math> are different sets of words that generate the same variety <math>\mathcal{V}</math>, the condition of being a homologism with respect to <math>W_1</math> coincides with the condition of being a homologism with respect to <math>W_2</math>.
 ==Related notions==
-* [[Isologism]] is a homomologism that is invertible, i.e., both its component homomorphisms are isomorphisms.
+{| class="sortable" border="1"
+! Term !! Meaning
+|-
+| [[category of groups with homologisms]] || this is a category whose objects are group and where the morphisms are homologisms of groups.
+|-
+| [[isologism of groups]] || this is a homologism where both the component homomorphisms are isomorphisms.
+|}
+==Particular cases==
+{| class="sortable" border="1"
+! Variety !! Generating word or set of words !! Corresponding notion of homologism
+|-
+| variety of [[abelian group]]s || <math>[x_1,x_2]</math> -- the [[commutator]] word || [[homoclinism of groups]]
+|-
+| variety of groups of nilpotency class at most <math>n</math> || <math>[[ \dots [[x_1,x_2],x_3],\dots,x_c],x_{c+1}]</math> -- the left-normed iterated commutator word || [[n-homoclinism of groups]]
+|}