No. |
Shorthand |
A subgroup of a group is a -Sylow subgroup if... |
A subgroup of a group is a -Sylow subgroup of if ... |
Applications to... |
Additional comments
|
1 |
order prime, index coprime |
it is a -group (i.e., its order is a power of , so it is a group of prime power order) and its index is relatively prime to |
is a -group (i.e., its order is for some nonnegative integer ), so it is a group of prime power order) and its index is relatively prime to . |
|
|
2 |
order is maximal power of prime definition |
the order of the subgroup is the largest power of dividing the order of the group |
the order of is the largest power of dividing the order of . In other words, if the order of is where is a nonnegative integer and is an integer relatively prime to , must have order . |
|
|
3 |
Hall subgroup plus p-subgroup definition |
it is a -group (nontrivial if divides the group order) and also a Hall subgroup: its order and index in the whole group are relatively prime |
it is a -group (nontrivial if divides the order of ) and also a Hall subgroup: its order and index are relatively prime. |
|
|
4 |
maximal element among p-subgroups |
it is a -subgroup of the whole group and is not contained in any bigger -subgroup |
is a -subgroup of (i.e., the order of is a power of ) and any -subgroup of containing must equal |
|
|
Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
|
order-dominating subgroup |
any subgroup of the whole group whose order divides the order of the Sylow subgroup is contained in a conjugate of the Sylow subgroup |
Sylow implies order-dominating |
|
|FULL LIST, MORE INFO
|
order-dominated subgroup |
any subgroup of the whole group whose order is a multiple of that of the Sylow subgroup contains a conjugate of the Sylow subgroup |
Sylow implies order-dominated |
|
|FULL LIST, MORE INFO
|
order-conjugate subgroup |
any other subgroup of the group of the same order is conjugate to it |
Sylow implies order-conjugate |
|
Order-dominated subgroup, Order-dominating subgroup|FULL LIST, MORE INFO
|
order-automorphic subgroup |
any other subgroup of the group of the same order is automorphic to it |
(also via order-conjugate) |
|
Order-conjugate subgroup|FULL LIST, MORE INFO
|
order-isomorphic subgroup |
any other subgroup of the group of the same order is isomorphic to it |
(via order-conjugate) |
|
Order-conjugate subgroup|FULL LIST, MORE INFO
|
isomorph-automorphic subgroup |
any other subgroup isomorphic to it is automorphic to it |
(via order-conjugate), see Sylow implies isomorph-automorphic |
|
Order-conjugate subgroup|FULL LIST, MORE INFO
|
isomorph-conjugate subgroup |
any other subgroup isomorphic to it is conjugate to it |
(via order-conjugate), see Sylow implies isomorph-conjugate |
|
Intermediately isomorph-conjugate subgroup, Nilpotent Hall subgroup, Order-conjugate subgroup, Order-dominated subgroup|FULL LIST, MORE INFO
|
automorph-conjugate subgroup |
any other subgroup automorphic to it is conjugate to it |
(via order-conjugate), see Sylow implies automorph-conjugate |
|
Characteristic subgroup of Sylow subgroup, Intermediately automorph-conjugate subgroup, Order-conjugate subgroup|FULL LIST, MORE INFO
|
intermediately isomorph-conjugate subgroup |
isomorph-conjugate in every intermediate subgroup |
Sylow implies intermediately isomorph-conjugate |
|
Intermediately isomorph-conjugate subgroup, Nilpotent Hall subgroup, Order-conjugate subgroup, Order-dominated subgroup|FULL LIST, MORE INFO
|
intermediately automorph-conjugate subgroup |
automorph-conjugate in every intermediate subgroup |
Sylow implies intermediately automorph-conjugate |
|
Intermediately isomorph-conjugate subgroup|FULL LIST, MORE INFO
|
Hall subgroup |
order and index are relatively prime |
(by definition) |
whole group gives counterexample when it isn't a -group |
Nilpotent Hall subgroup, Pronormal Hall subgroup|FULL LIST, MORE INFO
|
nilpotent Hall subgroup |
Hall and also nilpotent |
prime power order implies nilpotent |
whole group gives counterexample when it is nilpotent and not a -group |
|FULL LIST, MORE INFO
|
Order-dominating Hall subgroup |
Hall and order-dominating subgroup |
(via order-dominating + Hall) |
|
|FULL LIST, MORE INFO
|
procharacteristic subgroup |
|
Sylow implies procharacteristic |
|
Nilpotent Hall subgroup|FULL LIST, MORE INFO
|
weakly procharacteristic subgroup |
|
(via procharacteristic) |
|
|FULL LIST, MORE INFO
|
paracharacteristic subgroup |
|
(via procharacteristic) |
|
Hall subgroup, Join of Sylow subgroups|FULL LIST, MORE INFO
|
polycharacteristic subgroup |
|
(via procharacteristic) |
|
Hall subgroup, Join of Sylow subgroups|FULL LIST, MORE INFO
|
weakly characteristic subgroup |
|
(via procharacteristic) |
|
|FULL LIST, MORE INFO
|
intermediately normal-to-characteristic subgroup |
characteristic subgroup inside any intermediate subgroup in which it is normal |
|
|
Hall subgroup, Intermediately automorph-conjugate subgroup, Intermediately isomorph-conjugate subgroup|FULL LIST, MORE INFO
|
pronormal subgroup |
any conjugate subgroup is conjugate to it in their join |
Sylow implies pronormal |
|
Intermediately isomorph-conjugate subgroup, Nilpotent Hall subgroup, Nilpotent pronormal subgroup, Sylow subgroup of normal subgroup|FULL LIST, MORE INFO
|
weakly pronormal subgroup |
|
(via pronormal) |
|
Intermediately automorph-conjugate subgroup, Intermediately isomorph-conjugate subgroup, Pronormal subgroup|FULL LIST, MORE INFO
|
paranormal subgroup |
|
(via pronormal) |
|
Hall subgroup, Intermediately isomorph-conjugate subgroup, Join of Sylow subgroups, Pronormal subgroup|FULL LIST, MORE INFO
|
polynormal subgroup |
|
(via pronormal) |
|
Hall subgroup, Intermediately automorph-conjugate subgroup, Intermediately isomorph-conjugate subgroup, Join of Sylow subgroups, Paranormal subgroup, Weakly pronormal subgroup|FULL LIST, MORE INFO
|
weakly normal subgroup |
|
(via pronormal) |
|
Paranormal subgroup|FULL LIST, MORE INFO
|
intermediately subnormal-to-normal subgroup |
|
(via pronormal) |
|
Hall subgroup, Intermediately automorph-conjugate subgroup, Intermediately isomorph-conjugate subgroup, Intermediately normal-to-characteristic subgroup, Paranormal subgroup, Polynormal subgroup, Pronormal subgroup, Weakly pronormal subgroup|FULL LIST, MORE INFO
|
subgroup with abnormal normalizer |
|
(via pronormal) |
|
Pronormal subgroup|FULL LIST, MORE INFO
|
subgroup with weakly abnormal normalizer |
|
(via weakly pronormal) |
|
|FULL LIST, MORE INFO
|
subgroup with self-normalizing normalizer |
|
(via pronormal, weakly pronormal) |
|
Pronormal subgroup, Weakly pronormal subgroup|FULL LIST, MORE INFO
|
Closure-characteristic subgroup |
normal closure is a characteristic subgroup |
(via automorph-conjugate, also via Hall), see also Sylow implies closure-characteristic |
|
Automorph-conjugate subgroup, Hall subgroup, Join of Sylow subgroups, Subgroup whose normal closure is fully invariant, Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
|
Core-characteristic subgroup |
normal core is a characteristic subgroup |
(via automorph-conjugate, also via Hall), see also Sylow implies core-characteristic |
|
Automorph-conjugate subgroup, Hall subgroup, Intersection of Sylow subgroups, Subgroup whose normal core is fully invariant|FULL LIST, MORE INFO
|
WNSCDIN-subgroup |
|
(via pronormal), see also Sylow implies WNSCDIN |
|
Pronormal subgroup|FULL LIST, MORE INFO
|
MWNSCDIN-subgroup |
|
(via pronormal), see also Sylow implies MWNSCDIN |
|
Pronormal subgroup|FULL LIST, MORE INFO
|
subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed |
|
(via WNSCDIN) |
|
Pronormal subgroup|FULL LIST, MORE INFO
|
intermediately normal-to-complemented subgroup |
complemented normal subgroup in its normalizer |
Schur-Zassenhaus theorem |
|
|FULL LIST, MORE INFO
|
intermediately central factor-to-direct factor |
direct factor in any intermediate subgroup in which it is a central factor |
|
|
Permuting transfer-closed central factor-to-direct factor, Permuting transfer-closed conjugacy-closed-to-retract|FULL LIST, MORE INFO
|
intermediately conjugacy-closed-to-retract |
|
Conjugacy-closed and Sylow implies retract |
|
Permuting transfer-closed conjugacy-closed-to-retract|FULL LIST, MORE INFO
|