The term 2-group may refer to either of the following:
* Most typical ... * Usage in higher category theory: See strict 2-group and 2-group ... ...
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# uses::Strict characteristicity is quotient-transitive # uses::Strict characteristicity is strongly join-closed ... ...
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systems on a concrete instance of the group (strict, not up to isomorphism of fusion systems) || 2
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| Total number of fusion systems up to isomorphism ... ...
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on a concrete instance of the group (strict, not up to isomorphism of ... !! Number of such fusion systems under strict counting !! Can the fusion ... ...
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on a concrete instance of the group (strict, not up to isomorphism of ... !! Number of such fusion systems under strict counting !! Can the fusion ... ...
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on a concrete instance of the group (strict, not up to isomorphism of ... !! Number of such fusion systems under strict counting !! Can the fusion ... ...
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class !! Number of saturated fusion systems (strict counting) !! Number of saturated fusion systems up to isomorphism !! Is the group resistant? In ... ...
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closed subgroup property || Yes || strict characteristicity is strongly ... join-closed subgroup property || Yes || strict characteristicity is strongly ... ...
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These aren't strict converses, but converse-type statements:
* Degree of irreducible representation is bounded by index of abelian subgroup ... ...
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* Strict characteristicity is commutator-closed
===Other facts about commutators and normal subgroups===
* Commutator of the whole group and a ... ...
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group over any other field. This is unique in a strict sense, not merely up to isomorphisms.
==Summary==
{| class="sortable" border ... ...
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\alpha) is 2 (note that the latter extension is strict because \mathbb{Q}(\alpha) can be viewed as a subfield of \R but \mathbb{Q}(\alpha,\beta) cannot ... ...
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| 4 || |C(K)| \le |C(G)| and inequality is strict if G is finite and H is nontrivial. || || || ||
|} ... ...
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, if and only if q is a prime number rather than a strict prime power.
|}
==Linear representation theory==
Linear representation theory of general ... ...
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are all strict inclusions as long as we are within the group. The length of this chain of subgroups is at most equal to the max-length of the ... ...
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Combine center is strictly characteristic with strict characteristicity is quotient-transitive || --|| inductive proof methods for the ascending series ... ...
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is strongly intersection-closed]], and strict characteristicity is strongly intersection-closed.
Given: H_i is a family of normal subgroups of ... ...
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| Strictly characteristic subgroup || surjective endomorphisms || Strict characteristicity is strongly join-closed
|-
| Injective endomorphism ... ...
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property with respect to surjective endomorphisms. Strict characteristicity is strongly intersection-closed
* Fully invariant subgroup: This is ... ...
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happen only if d = 1, hence for d > 1, we have strict inequality. Finally, each |q - \zeta| > 1 because q \ge 2 so q - 1 \ge 1. The upshot is ... ...
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