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A Hall retract of a finite group is a Hall subgroup that is also a ... * Stronger than::Order-conjugate subgroup * Stronger than::Order-isomorphic ... ...

Any Sylow subgroup of a finite group is an isomorph-conjugate subgroup: ... * Sylow implies intermediately isomorph-conjugate * Sylow implies automorph ... ...

Any fact about::nilpotent Hall subgroup (i.e., a fact about::Hall ... # uses::Sylow implies intermediately isomorph-conjugate ... ...

An intermediately characteristic subgroup of an intermediately characteristic ... The cyclic subgroup generated by a is a subgroup of order four, and ... ...

A subgroup H of a finite group G is termed a pronormal Hall subgroup ... # H is a Hall subgroup of G: The order and index of H in G are relatively ... ...

A subgroup of a group is termed isomorph-conjugate if any subgroup ... A subgroup H of a group G is termed isomorph-conjugate if whenever ... ...

* Hall and order-conjugate not implies order-dominating * Hall and order-conjugate not implies order-dominated * Hall not implies ... ...

* Suppose P is a p-Sylow subgroup of a finite group G, and Q is a ... that the condition of being a p-Sylow subgroup is completely determined ... ...

A subgroup of a finite group is termed a nilpotent Hall subgroup if ... * Weaker than::Abelian Hall subgroup * Weaker than::Sylow subgroup ... ...

| order of the subgroup || 16 || |- | index of the subgroup || 3 || |- | size of conjugacy class of subgroups ... ...

Consider the subgroup: H = H_5 := \{ (), (1,2)(3,4), (1,3)(2,4), (1 ... This subgroup is isomorphic to the Klein four-group. There are five ... ...

It is possible to have an abelian group G and an intermediately characteristic ... * Equivalence of definitions of intermediately characteristic subgroup ... ...

Consider the subgroup: H = H_{\{1,2,3 \}} = \{ (), (1,2,3), (1,3,2) \} of size two is fixed point-wise by that conjugate subgroup: ... ...

A Hall subgroup of a group need not be conjugate to all its automorphs. ... * Hall not implies intermediately automorph-conjugate * Hall not implies ... ...

of maximum order in P, i.e., J(P) is the subgroup generated by all abelian ... * Join of abelian subgroups of maximum order in Sylow subgroup is ... ...

A subgroup of a finite group is termed a Sylow subgroup if it is a ... ! No. !! Shorthand !! A subgroup of a group is a p-Sylow subgroup ... ...

A subgroup H of a group G is said to be isomorph-free if it satisfies ... H \cong K, then H = K (i.e. H and K are the same subgroup). ... ...

* Existence: For any prime p, there exists a p-Sylow subgroup ... * Domination: Any p-subgroup is contained inside some p-Sylow subgroup ... ...

need not be characteristic in every intermediate subgroup. ... x, the center \{ e, a^2 \} is not intermediately characteristic. In fact ... ...

A subgroup of a group is said to be intermediately characteristic ... A subgroup H of a group G is said to be intermediately characteristic ... ...