ZJ-subgroup contains D*-subgroup
This article gives the statement, and possibly proof, of the subgroup obtained from one subgroup-defining function, namely D*-subgroup, always being contained in the subgroup obtained from another subgroup-defining function, namely ZJ-subgroup.
View other such results
Denote by the set:
Denote by the set of abelian subgroups of maximum order in . Then, the ZJ-subgroup is defined as the intersection of all members of . Equivalently, it is the center of the join of all these subgroups (see join of abelian subgroups of maximum order).
- Characteristic implies normal
- Stable version of Thompson's replacement theorem for abelian subgroups
- Group generated by finitely many abelian normal subgroups is nilpotent of class at most equal to the number of subgroups
- Nilpotency of fixed class is subgroup-closed
- Nilpotent implies no proper contranormal subgroup and prime power order implies nilpotent
It suffices to show that is contained in every abelian subgroup of maximum order in , because is the intersection of all these.
Given: A finite -group , , is an abelian subgroup of maximum order in .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||is an abelian characteristic subgroup, hence abelian normal subgroup of .||Fact (1)||is the D*-subgroup||abelianness follows from containment in , and characteristic follows from it being the unique maximal element.|
|2||The product is a subgroup of that is metabelian.||Step (1)||[SHOW MORE]|
|3||is an abelian subgroup of maximum order in .||is abelian of maximum order in||Given-direct|
|4||There exists a subgroup of that is abelian and normalized by , is contained in the normal closure , and has the same order as (and hence maximum order among abelian subgroups of ). In particular, normalizes .||Fact (2)||Apply Fact (2) in the ambient group .|
|5||has nilpotency class at most two.||Fact (3)||Steps (1), (4)||Apply Fact (3) and Steps (1) and (4) inside the ambient group . Note here that is not necessarily normal, but since it is normalized by , it is normal in the join , which is the group within which we apply Fact (3).|
|6||For any , the subgroup is a subgroup of of class at most two.||Fact (4)||Step (5)||Fact-step combination direct|
|7||For any , centralizes .||and hence .||Step (6)||Apply Step (6) and the condition that to get the result.|
|8||The product is an abelian subgroup of||Steps (1), (4), (7)||By Steps (1), (4), both are abelian. By Step (7), they centralize each other. Hence their product is also abelian.|
|9||, so .||Steps (4), (8)||is a subgroup of that contains . By Step (4), is abelian of maximum order in . This forces .|
|10||.||Steps (4), (9)||, and by Step (4), .|
|11||, so is a contranormal subgroup of .||Step (10)||By the previous step, . The other direction, , is definitional.|
|12||, so .||Fact (5)||we are working with -groups||Step (11)||Step-fact-given combination direct|
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format