Sentence examples similar to their proper subgroups from inspiring English sources

The subsequent work that resulted in Thompson's receiving the Fields Medal was the determination of all the minimal simple finite groups that is, those groups all of whose proper subgroups are built only of cyclic composition factors.

It shows, in particular, that (mathrm{SO}_4({mathbb {R}})) has no proper subgroups of dimension at least 5 and all subgroups of dimension 4 are conjugate in (mathrm{O}_4({mathbb {R}})) to the unitary group (mathrm{U}_2subset mathrm{mathbb{R}}hbb {R}})).

Suppose that (p) is a prime and (G) is a minimal non- p -nilpotent group, i.e., (G) is non- p -nilpotentnon- p -nilpotente proper subgroups are all (p)-nilpotent.e

Then (HK_{0}) is a proper subgroup of (G).

Clearly, (mathcal {H}) is a proper subgroup because (r+r^{21} =jOmega.) Let (gamma ) and (gamma ') satisfy (5.2).

and Sp ( N / 2 ) is a proper subgroup of SU ( N ).

A group G is minimal nonsolvable if G is nonsolvable but every proper subgroup of G is solvable.

It is a proper subgroup, so that by Proposition 5.5 P ′ contains a copy of G m, R.

In this case, the Weyl group of the associated Lie algebra is isomorphic to a proper subgroup of the isotropy group of the Weyl groupoid.

In view of Z ( G ) ⩽ C G ( x ), one has that Z ( G ) is a proper subgroup of G, and 3, 7 ∉ π ( Z ( G ) ).

Therefore, F ∗ ( E ) is a proper subgroup of E. (2) Let p be the smallest prime dividing the order of F and P be a Sylow p-subgroup of F.