Suppose G is a finite simple group and P is a p-Sylow subgroup of G. Consider the fusion system induced by G on P. This must be a simple fusion ... ...
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given group can be embedded between the simple group and its automorphism group. ... * Almost simple not implies simple or complete: An almost simple group ... ...
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simple group operator|normal subgroup simple is obtained by applying the simple group operator to the subgroup ... ...
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A group is termed an infinite simple group if it satisfies the following ... or, its order is infinite) as well as a simple group (it has no proper nontrivial ... ...
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#redirect simple periodic group ... ...
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Equivalence of definitions of characteristically simple group ... simple group operator|characteristic subgroup
The group property of ... ...
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A topologically simple group is a defining ingredient::topological ... In particular, for a topologically simple group, it must be either the ... ...
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===In terms of the simple group operator=== simple is obtained by applying the simple group operator to the trim subgroup ... ...
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A sporadic simple group is a finite simple group that is not an alternating group, classical group, or exceptional group of Lie type. ... ...
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#redirect splitting-simple group ... ...
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The simple group operator is an operator from the collection of trim ... * Normal subgroup gives rise to simple group
* Characteristic subgroup ... ...
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==Definition==
===Symbol-free definition===
A group (possibly with additional structures and relations) is said to be definably simple if it has ... ...
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A group is said to be a finite simple group if it satisfies both these ... * It is a simple group, viz it has no proper nontrivial normal subgroup ... ...
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A group is said to be a minimal simple group if it is a simple non-abelian group and every proper subgroup of it is solvable.
==Relation with ... ...
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A simple group of Ree type is a finite simple non-Abelian group satisfying the following conditions:
* It has an Abelian Sylow 2-subgroup ... ...
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| Weaker than::simple group || nontrivial and no proper nontrivial ... group is simple || || || splitting-simple group|quasisimple group ... ...
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finitely generated group|simple group
==Definition== A finitely generated simple group is a nontrivial group that is both ... ...
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A locally finite simple group is a group that is both a locally finite ... * Every finite simple group is a locally finite simple group. ... ...
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A finite simple non-Abelian group G is said to be a simple group of component type if for any involution t, the group C_G(t)/O(C_G(t)) has no ... ...
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A finitely presented simple group is a group that is both a finitely ... finitely presented group|simple group
==Relation with other properties== ... ...
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old property = simple group|
new generic context = Lie ring|
new specific context = Lie ring}}
==Definition==
A Lie ring is said to be simple ... ...
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there exists a fact about::minimal simple group arising as a subquotient of G. ... To prove: There exists a minimal simple group arising as a subquotient ... ...
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old property = simple group|
new generic context = IAPS|
new specific context = IAPS}}
==Definition==
===Symbol-free definition=== ... ...
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A simple group of Ree type is a finite simple non-Abelian group satisfying the following conditions:
* It has an Abelian Sylow 2-subgroup ... ...
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The simple group operator is an operator from the collection of trim ... * Normal subgroup gives rise to simple group
* Characteristic subgroup ... ...
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simple group|periodic group
==Definition== called simple torsion group or periodic simple group or torsion simple group ... ...
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A sporadic simple group is a finite simple group that is not an alternating group, classical group, or exceptional group of Lie type. ... ...
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===In terms of the simple group operator=== simple is obtained by applying the simple group operator to the trim subgroup ... ...
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given group can be embedded between the simple group and its automorphism group. ... * Almost simple not implies simple or complete: An almost simple group ... ...
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property = simple group|
metaproperty = directed union-closed group ... ===Simple group===
A nontrivial group is simple if it has no proper ... ...
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on S under composition, is a simple group.
==Facts used==
# uses::Alternating groups are simple: The alternating group on a finite set is simple ... ...
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A finite simple non-Abelian group G is said to be a simple group of component type if for any involution t, the group C_G(t)/O(C_G(t)) has no ... ...
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Then the symmetric group on S is an almost simple group. ... ===Almost simple group===
A group G is termed an almost simple group ... ...
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A locally finite simple group is a group that is both a locally finite ... * Every finite simple group is a locally finite simple group. ... ...
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A group is termed an infinite simple group if it satisfies the following ... or, its order is infinite) as well as a simple group (it has no proper nontrivial ... ...
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normal subgroup|simple group
==Definition==
===Symbol-free definition===
A subgroup of a group is termed a simple subnormal subgroup or a minimal ... ...
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for the situation of a finite simple group, where the prime equals 2 ... ...
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A simple group need not be a co-Hopfian group: it is possible to have ... subgroup H. by fact (1), we thus have a simple group that is isomorphic to ... ...
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finitely generated group|simple group
==Definition== A finitely generated simple group is a nontrivial group that is both ... ...
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A group is said to be a finite simple group if it satisfies both these ... * It is a simple group, viz it has no proper nontrivial normal subgroup ... ...
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