Four characterization results

Examples show that Rédei’s theorem does not hold for normalized factorizations with general factors. For instance let G be the cyclic group of order 8 with generator x and let <math display='block'> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>=</mo><mo>&#x007B;</mo><mi>e</mi><mo>,</mo><msup> <mi>x</mi> <mn>2</mn> </msup> <mo>&#x007D;</mo><mo>,</mo><mtext>&#x2003;</mtext><msub> <mi>A</mi> <mn>2</mn> </msub> <mo>=</mo><mo>&#x007B;</mo><mi>e</mi><mo>,</mo><mi>x</mi><mo>,</mo><msup> <mi>x</mi> <mn>4</mn> </msup> <mo>,</mo><msup> <mi>x</mi> <mn>5</mn> </msup> <mo>&#x007D;</mo><mo>.</mo> </math> $${A_1} = \{ e,{x^2}\} ,\quad {A_2} = \{ e,x,{x^4},{x^5}\} $$. Now G = A₁A₂ is a normalized factorization of G and the factors A₁ and A₂ are not subgroups of G.