| Function |
Value |
Similar groups |
Explanation
|
| underlying prime of p-group |
{{{underlying prime}}} |
|
|
| order (number of elements, equivalently, cardinality or size of underlying set) |
{{{order}}} |
groups with same order"{{{order}}}" is not a number. |
|
| prime-base logarithm of order |
{{{order p-log}}} |
groups with same prime-base logarithm of order"{{{order p-log}}}" is not a number. |
|
| max-length of a group |
{{{order p-log}}} |
|
max-length of a group equals prime-base logarithm of order for group of prime power order
|
| chief length |
{{{order p-log}}} |
|
chief length equals prime-base logarithm of order for group of prime power order
|
| composition length |
{{{order p-log}}} |
|
composition length equals prime-base logarithm of order for group of prime power order
|
| exponent of a group |
{{{exponent}}} |
groups with same order and exponent of a group<ul><li>"{{{exponent}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and exponent of a group<ul><li>"{{{exponent}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same exponent of a group"{{{exponent}}}" is not a number. |
|
| prime-base logarithm of exponent |
{{{exponent p-log}}} |
groups with same order and prime-base logarithm of exponent<ul><li>"{{{exponent p-log}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and prime-base logarithm of exponent<ul><li>"{{{exponent p-log}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same prime-base logarithm of exponent"{{{exponent p-log}}}" is not a number. |
|
| Frattini length |
{{{exponent p-log}}} |
groups with same order and Frattini length<ul><li>"{{{exponent p-log}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and Frattini length<ul><li>"{{{exponent p-log}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same Frattini length"{{{exponent p-log}}}" is not a number. |
Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
|
| minimum size of generating set |
{{{rank}}} |
groups with same order and minimum size of generating set<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and minimum size of generating set<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same minimum size of generating set"{{{rank}}}" is not a number. |
|
| subgroup rank of a group |
{{{rank}}} |
groups with same order and subgroup rank of a group<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and subgroup rank of a group<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same subgroup rank of a group"{{{rank}}}" is not a number. |
same as minimum size of generating set since it is an abelian group of prime power order
|
| rank of a p-group |
{{{rank}}} |
groups with same order and rank of a p-group<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and rank of a p-group<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same rank of a p-group"{{{rank}}}" is not a number. |
same as minimum size of generating set since it is an abelian group of prime power order
|
| normal rank of a p-group |
{{{rank}}} |
groups with same order and normal rank of a p-group<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and normal rank of a p-group<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same normal rank of a p-group"{{{rank}}}" is not a number. |
same as minimum size of generating set since it is an abelian group of prime power order
|
| characteristic rank of a p-group |
{{{rank}}} |
groups with same order and characteristic rank of a p-group<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same prime-base logarithm of order and characteristic rank of a p-group<ul><li>"{{{rank}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same characteristic rank of a p-group"{{{rank}}}" is not a number. |
same as minimum size of generating set since it is an abelian group of prime power order
|
| nilpotency class |
1 |
|
The group is a nontrivial abelian group
|
| derived length |
1 |
|
The group is a nontrivial abelian group
|
| Fitting length |
1 |
|
The group is a nontrivial abelian group
|