Isomorphic iff potentially conjugate: Difference between revisions

Latest revision as of 02:44, 30 March 2013

Statement

For just one pair of isomorphic subgroups

Suppose $G$ is a group and $H, K \leq G$ are isomorphic groups, i.e., there is an isomorphism of groups, say $σ$ , from $H$ to $K$ (Note that this isomorphism need not arise from an automorphism of $G$ , so $H$ and $K$ need not be automorphic subgroups).

Then, there exists a group $L$ containing $G$ such that $H, K$ are conjugate subgroups inside $L$ , and the induced isomorphism from $H$ to $K$ by that conjugating element equals $σ$ .

For a collection of many pairs of isomorphisms between subgroups

Suppose $G$ is a group, $I$ is an indexing set, and $H_{i} ≅ K_{i}$ are pairs of isomorphic subgroups of $G$ for each $i \in I$ . Let $σ_{i} : H_{i} \to K_{i}$ be an isomorphism for each $i \in I$ .

Then, there exists a group $L$ containing $G$ as a subgroup such that $H_{i}$ and $K_{i}$ are conjugate subgroups in $L$ for each $i \in I$ . More specifically, we can find $g_{i}, i \in I$ such that the map induced by conjugation by $g_{i}$ induces the isomorphism $σ_{i}$ .

Moreover, there is a natural construction of such a group $L$ , called a HNN-extension. In the case that $G$ is an torsion-free group, we can ensure that the group $L$ is also torsion-free.

Related facts

For finite groups

Isomorphic iff potentially conjugate in finite: This construction works when the original group is finite and yields a bigger group that is also finite.

Facts about automorphisms extending to inner automorphisms

Inner automorphism to automorphism is right tight for normality: In other words, if $σ$ is an automorphism of $G$ , there exists a group $L$ containing $G$ as a normal subgroup, and an inner automorphism of $L$ whose restriction to $G$ equals $σ$ .
Left transiter of normal is characteristic: A direct application of the fact that any automorphism of a group extends to an inner automorphism in a bigger group containing it as a normal subgroup. This says that $H \leq G$ is such that (whenever $G$ is normal in $L$ , $H$ is also normal in $L$ ) if and only if $H$ is characteristic in $G$ .
Characteristic of normal implies normal

Facts about injective endomorphisms

Every injective endomorphism arises as the restriction of an inner automorphism

Applications

Same order iff potentially conjugate: $x, y \in G$ are such that $x, y$ have the same order if and only if then there is a group $L$ containing $G$ in which $x$ and $y$ are conjugate elements. This is a direct application based on looking at the cyclic subgroups $⟨ x ⟩$ and $⟨ y ⟩$ .
Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes
Every torsion-free group is a subgroup of a simple torsion-free group

@@ Line 3: / Line 3: @@
 ===For just one pair of isomorphic subgroups===
-Suppose <math>G</math> is a group and <math>H, K \le G</math> are [[isomorphic subgroups]], i.e., there is an [[isomorphism of groups]] from <math>H</math> to <math>K</math> (Note that this isomorphism need ''not'' arise from an automorphism of <math>G</math>, so <math>H</math> and <math>K</math> need not be [[automorphic subgroups]]). The,n there exists a group <math>L</math> containing <math>G</math> such that <math>H, K</math> are [[conjugate subgroups]] inside <math>L</math>.
+Suppose <math>G</math> is a group and <math>H, K \le G</math> are [[isomorphic groups]], i.e., there is an [[isomorphism of groups]], say <math>\sigma</math>, from <math>H</math> to <math>K</math> (Note that this isomorphism need ''not'' arise from an automorphism of <math>G</math>, so <math>H</math> and <math>K</math> need not be [[automorphic subgroups]]).
-===For a collection of many pairs of isomorphism subgroups===
+Then, there exists a group <math>L</math> containing <math>G</math> such that <math>H, K</math> are [[conjugate subgroups]] inside <math>L</math>, and the induced isomorphism from <math>H</math> to <math>K</math> by that conjugating element equals <math>\sigma</math>.
-Suppose <math>G</math> is a group, <math>I</math> is an indexing set, and <math>H_i \cong K_i</math> are pairs of isomorphic subgroups of <math>G</math> for each <math>i \in I</math>. Then, there exists a group <math>L</math> containing <math>G</math> as a subgroup such that <math>H_i</math> and <math>K_i</math> are [[conjugate subgroups]] in <math>L</math> for each <math>i \in I</math>. (Note: The choice of conjugating element may differ for different <math>i \in I</math>).
-Moreover, there is a ''natural'' construction of such a group <math>L</math>, called a [[HNN-extension]]. In the case that <math>G</math> is a [[torsion-free group]], we can ensure
+===For a collection of many pairs of isomorphisms between subgroups===
+Suppose <math>G</math> is a group, <math>I</math> is an indexing set, and <math>H_i \cong K_i</math> are pairs of isomorphic subgroups of <math>G</math> for each <math>i \in I</math>. Let <math>\sigma_i: H_i \to K_i</math> be an isomorphism for each <math>i \in I</math>.
+Then, there exists a group <math>L</math> containing <math>G</math> as a subgroup such that <math>H_i</math> and <math>K_i</math> are [[conjugate subgroups]] in <math>L</math> for each <math>i \in I</math>. More specifically, we can find <math>g_i, i \in I</math> such that the map induced by conjugation by <math>g_i</math> induces the isomorphism <math>\sigma_i</math>.
+Moreover, there is a ''natural'' construction of such a group <math>L</math>, called a [[HNN-extension]]. In the case that <math>G</math> is an [[torsion-free group]], we can ensure that the group <math>L</math> is also torsion-free.
 ==Related facts==
+===For finite groups===
+* [[Isomorphic iff potentially conjugate in finite]]: This construction works when the original group is finite and yields a bigger group that is also finite.
 ===Facts about automorphisms extending to inner automorphisms===
@@ Line 19: / Line 28: @@
 * [[Characteristic of normal implies normal]]
+===Facts about injective endomorphisms===
+* [[Every injective endomorphism arises as the restriction of an inner automorphism]]
 ===Applications===
 * [[Same order iff potentially conjugate]]: <math>x,y \in G</math> are such that <math>x,y</math> have the same order if and only if then there is a group <math>L</math> containing <math>G</math> in which <math>x</math> and <math>y</math> are conjugate elements. This is a direct application based on looking at the cyclic subgroups <math>\langle x \rangle</math> and <math>\langle y \rangle</math>.
-* [[Every group is a subgroup of a group with two conjugacy classes]]
-* [[Every group is a subgroup of a simple group]]
 * [[Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes]]
+* [[Every torsion-free group is a subgroup of a simple torsion-free group]]