# ZJ-subgroup contains D*-subgroup

From Groupprops

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.

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## Statement

Suppose is a prime number and is a finite p-group. Then, the ZJ-subgroup of contains the D*-subgroup of .

## Definitions used

### D*-subgroup

Denote by the set:

### ZJ-subgroup

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).

## Facts used

- 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

## Proof

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 .

**To prove**:

**Proof**:

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 |

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