It is possible to have a group G, a fact about::4-subnormal subgroup H of G, and a fact about::finite subnormal subgroup K of G, such that the ... ...
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#REDIRECT Subnormality is normalizing join-closed ... ...
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Suppose B is a group and A is a fact about::3-subnormal subgroup of ... * There is no unique largest fact about::2-subnormal subgroup of B ... ...
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Suppose k is a positive integer. Then, we can find a 2-subnormal subgroup ... not implies normal: We can have a 2-subnormal subgroup that is not normal ... ...
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Suppose H is a subnormal subgroup of G of subnormal depth n; in other ... This result shows that for any subnormal subgroup, there is a unique ... ...
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Suppose H is a fact about::2-subnormal subgroup of a group G. Then ... descending subnormal series for any subnormal subgroup H \le G -- this is ... ...
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A group in which every subnormal subgroup is 2-subnormal is a group ... # Every subnormal subgroup is a 2-subnormal subgroup, i.e., its defining ... ...
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Suppose G is a group, H is a 2-subnormal subgroup and A is a finite ... is a subnormal subgroup of G.
==Related facts==
* Commutator of a group ... ...
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Suppose G is a group, H is a 2-subnormal subgroup and A is a subset ... is a 3-subnormal subgroup of G.
==Related facts==
* Commutator of a ... ...
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A subgroup of a group is termed a subnormal-permutable subnormal subgroup ... * Weaker than::Perfect subnormal subgroup
===Weaker properties=== ... ...
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The notion of c-subnormal subgroup was considered by Jaraden Jehad in a paper in 2000. Later, an equivalent notion of weakly c-normal subgroup ... ...
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A subgroup of a group is termed a 4-subnormal subgroup if it satisfies ... * It is a defining ingredient::subnormal subgroup of defining ingredient::subnormal ... ...
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A subgroup H of a group G is termed an almost subnormal subgroup ... | Weaker than::Normal subgroup || || || || almost subnormal subgroup ... ...
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A subgroup of a group is termed a 3-subnormal subgroup if it satisfies ... # It is a subnormal subgroup and its subnormal depth is at most three. ... ...
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is termed the subnormal depth of the subnormal subgroup. ... A k-subnormal subgroup is a subnormal subgroup with subnormal depth ... ...
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subnormal subgroup|nilpotent group
==Definition== A subgroup of a group is termed a nilpotent subnormal subgroup if ... ...
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K of G, H is a defining ingredient::subnormal subgroup of \langle H, K \rangle. ... locally subnormal || || locally subnormal subgroup|permutable subgroup ... ...
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A subgroup of a group is termed a finite subnormal subgroup if it ... * Subnormal subgroup of finite index
==Facts==
* Join of two finite ... ...
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#redirect simple subnormal subgroup ... ...
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A subgroup of a group is termed a simple subnormal subgroup or a minimal ... * Subnormal subgroup
==Metaproperties==
In fact, the only simple subnormal ... ...
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stronger = permutable subnormal subgroup|
weaker = join-transitively ... such that H is a permutable subnormal subgroup -- it is both subnormal ... ...
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Suppose G is a group, H is a 3-subnormal subgroup of G and K is a ... subnormal: In fact, the join of a 2-subnormal subgroup and a s-subnormal subgroup ... ...
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stronger = perfect subnormal subgroup|
weaker = join-transitively subnormal ... A perfect subnormal subgroup of a group is join-transitively subnormal: ... ...
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Suppose k is a positive integer. Then, we can find a 2-subnormal subgroup ... not implies normal: We can have a 2-subnormal subgroup that is not normal ... ...
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subnormal: The join of a 2-subnormal subgroup and a k-subnormal subgroup ... A join of a normal subgroup and a 2-subnormal subgroup is 2-subnormal. More ... ...
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stronger = perfect subnormal subgroup|
weaker = subnormal-permutable ... Any perfect subnormal subgroup of a group is subnormal-permutable: ... ...
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A 2-subnormal subgroup of a 2-subnormal subgroup need not be 2-subnormal. ... ===Left-transitively 2-subnormal subgroup===
Left-transitively 2-subnormal ... ...
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property = subnormal subgroup|
metaproperty = intersection-closed subgroup property}}
==Statement==
An arbitrary intersection of subnormal subgroups ... ...
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A subgroup of a group is termed a 4-subnormal subgroup if it satisfies ... * It is a defining ingredient::subnormal subgroup of defining ingredient::subnormal ... ...
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A subgroup of a group is termed a subnormal-permutable subnormal subgroup ... * Weaker than::Perfect subnormal subgroup
===Weaker properties=== ... ...
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A subgroup of a group is termed a 3-subnormal subgroup if it satisfies ... # It is a subnormal subgroup and its subnormal depth is at most three. ... ...
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left = join-transitively subnormal subgroup|
right = normal subgroup| ... of G and H is a join-transitively subnormal subgroup of K (in other words ... ...
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Any 3-subnormal subgroup of a group is finite-conjugate-join-closed ... (1) and (2), and the fact that a 2-subnormal subgroup of a normal subgroup ... ...
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Suppose H is a subnormal subgroup of G of subnormal depth n; in other ... This result shows that for any subnormal subgroup, there is a unique ... ...
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A subgroup of a group is termed a permutable subnormal subgroup if ... subgroup of finite group is a permutable subnormal subgroup. ... ...
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The property of being a subnormal subgroup is finite-relative-intersection ... Suppose H, K are subgroups of G such that H is a subnormal subgroup ... ...
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subgroups) is a T-group: every subnormal subgroup of it is normal. ... A group is termed a T-group if every subnormal subgroup of the group ... ...
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# Any normal subgroup of a group is a join-transitively subnormal subgroup. # The join of a normal subgroup and a subnormal subgroup is subnormal. ... ...
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The proof follows from the observation that a proper subnormal subgroup is normal in a strictly bigger subgroup (namely, the one adjacent to ... ...
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A subgroup of a group is termed a simple subnormal subgroup or a minimal ... * Subnormal subgroup
==Metaproperties==
In fact, the only simple subnormal ... ...
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