Qué $A^{-1}A=G$ implica que $AA^{-1}=G$?

Question

Deje $G$ ser un grupo, $A\subseteq G$ y poner $A^{-1}=\{ a^{-1}:a\in A\}$.

Es cierto que si $A^{-1}A=G$ $AA^{-1}=G$ (y viceversa)?

Answer 1

Creo que podría construir un azar contra-ejemplo con GAP (s es el subconjunto $A$ , $t$ es el conjunto de los productos de $A^{-1}A$ formación $G$ , $w$ es el conjunto de los productos de $AA^{-1}$)

gap> s:=[ (), (3,4), (2,4,3), (2,4), (1,2), (1,2,3), (1,2,3,4), (1,3,2,4), (1,4,2,3) ];t:=List([]); w:=List([]); for u in s do for v in s do t:=Union(t,List([u^(-1)*v])); w:=Union(w,List([u*v^(-1)])); od;od; Print(s,"\n");Print(t,"\n");Print(w,"\n");Print(StructureDescription(AsGroup(t)),"\n");Print(AsGroup(w),"\n");
[ (), (3,4), (2,4,3), (2,4), (1,2), (1,2,3), (1,2,3,4), (1,3,2,4), (1,4,2,3) ]
[  ]
[  ]
[ (), (3,4), (2,4,3), (2,4), (1,2), (1,2,3), (1,2,3,4), (1,3,2,4),   (1,4,2,3) ]
[ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4),
  (1,3,2), (1,3,4,2), (1,3), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3),
  (1,4)(2,3) ]
[ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4),
  (1,3,2), (1,3,4,2), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3) ]
S4
fail
gap>

El subconjunto $A$ es

$$[ (), (3,4), (2,4,3), (2,4), (1,2), (1,2,3), (1,2,3,4), (1,3,2,4), (1,4,2,3)]$$

The products forming the group $G$ are :

$$[ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3) ] $$

The permutations of the product $AA^{-1}$ are

$$[ (), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3) ]$$

$Un$ is a subset of $G$, as required. The permutation $(13)$ es la que falta en el último set.