1-isomorphic groups: Difference between revisions

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This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Suppose $G_{1}$ and $G_{2}$ are groups. We say that $G_{1}$ and $G_{2}$ are 1-isomorphic if there exists a 1-isomorphism between $G_{1}$ and $G_{2}$ , i.e., a 1-homomorphism of groups from $G_{1}$ to $G_{2}$ whose inverse is also a 1-homomorphism. In other words, there is a bijection between $G_{1}$ and $G_{2}$ whose restriction to any cyclic subgroup on either side is an isomorphism to its image.

Historical term

G. A. Miller used the term conformal groups to describe what are referred to here are 1-isomorphic groups. However, the term "conformal group" has a different, much more famous meaning as the group of angle-preserving symmetries of a geometric space.

Finite version

Two finite groups that are 1-isomorphic are termed 1-isomorphic finite groups. There are many equivalent characterizations of 1-isomorphic finite groups.

Facts

Any Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring. This shows that many groups of small prime power order are 1-isomorphic to abelian groups. Further information: Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring

Relation with other relations

Stronger relations

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorphic groups				\|FULL LIST, MORE INFO

Weaker relations

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
directed power graph-equivalent groups				\|FULL LIST, MORE INFO
undirected power graph-equivalent groups				\|FULL LIST, MORE INFO

@@ Line 4: / Line 4: @@
 ==Definition==
-Suppose <math>G_1</math> and <math>G_2</math> are [[group]]s. We say that <math>G_1</math> and <math>G_2</math> are '''1-isomorphic''' if there exists a [[1-isomorphism of groups|1-isomorphism]] between them, i.e., a [[1-homomorphism of groups]] from <math>G_1</math> to <math>G_2</math> whose inverse is also a 1-homomorphism. In other words, there is a bijection between <math>G_1</math> and <math>G_2</math> whose restriction to any cyclic subgroup on either side is an isomorphism to its image.
+Suppose <math>G_1</math> and <math>G_2</math> are [[group]]s. We say that <math>G_1</math> and <math>G_2</math> are '''1-isomorphic''' if there exists a [[1-isomorphism of groups|1-isomorphism]] between <math>G_1</math> and <math>G_2</math>, i.e., a [[1-homomorphism of groups]] from <math>G_1</math> to <math>G_2</math> whose inverse is also a 1-homomorphism. In other words, there is a bijection between <math>G_1</math> and <math>G_2</math> whose restriction to any cyclic subgroup on either side is an isomorphism to its image.
-==Relation with other properties==
+===Historical term===
-===Stronger properties===
+G. A. Miller used the term ''conformal groups'' to describe what are referred to here are 1-isomorphic groups. However, the term "conformal group" has a different, much more famous meaning as the group of angle-preserving symmetries of a geometric space.
+==Finite version==
+Two finite groups that are 1-isomorphic are termed [[1-isomorphic finite groups]]. There are many equivalent characterizations of 1-isomorphic finite groups.
+==Facts==
 Any [[Lazard Lie group]] is 1-isomorphic to the additive group of its Lazard Lie ring. This shows that many [[group of prime power order|groups of small prime power order]] are 1-isomorphic to [[abelian group of prime power order|abelian groups]]. {{further|[[Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring]]}}
-===Weaker properties===
-* For finite groups, [[order statistics-equivalent finite groups]]: {{proofofstrictimplicationat|[[1-isomorphic implies order statistics-equivalent]]|[[order statistics-equivalent not implies 1-isomorphic]]}}
+==Relation with other relations==
+===Stronger relations===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::isomorphic groups]] || || || || {{intermediate notions short|1-isomorphic groups|isomorphic groups}}
+|}
+===Weaker relations===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::directed power graph-equivalent groups]] || || || || {{intermediate notions short|directed power graph-equivalent groups|1-isomorphic groups}}
+|-
+| [[Stronger than::undirected power graph-equivalent groups]] || || || || {{intermediate notions short|undirected power graph-equivalent groups|1-isomorphic groups}}
+|}
+See also [[1-isomorphic finite groups#Weaker properties]].