Statement:
If \(p\) is a prime and if \(\{G_y: y\in Y\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y}G_y\) has a maximal \(p\)-subgroup.
Howard_Rubin_Number: 308-p
Parameter(s): This form depends on the following parameter(s): \(q\), \(q\): prime number
This form's transferability is: Unknown
This form's negation transferability is: Negation Transferable
Article Citations:
Howard-Yorke-1987: Maximal p-subgroups and the axiom of choice
Book references
Note connections:
Note 24
This note contains some definitions from group
theory that are used in \(\cal M22\), and forms [62 C], [62 D], [67 D], Form 180, and Form 308\((p)\)
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 308 A-p | If \(p\) is a prime and if \(Y\) is a set of non-empty, finite sets, then the weak direct product \(\prod_{y\in Y} S_y\) has a maximal \(p\)-subgroup. (\(S_y\) is the symmetric group on \(y\).) |
Howard-Yorke-1987
Note [24] |
| 308 B-p | If \(p\) is a prime and if \(Y\) is a set of non-empty finite sets such that \((\forall y\in Y)(\)gcd\((|y|,p)=1)\), then \(Y\) has a choice function. (This is similar to Form 46(\(K\)), but in Form 46(\(K\)) the set \(K\) is required to be finite.) |
Howard-Yorke-1987
|
| 308 C-p | If \(p\) is a prime and if \(Y\) is a set of non-empty finite sets such that \((\forall y\in Y)(|y|\equiv 1\mod p)\), then \(Y\) has a choice function. (This is similar to Form 46(\(K\)), but in Form 46(\(K\)) the set \(K\) is required to be finite.) |
Howard-Yorke-1987
|