| Function |
Value |
Similar groups |
Explanation for function value
|
| underlying prime of p-group |
2 |
|
|
| 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 |
{{{degree}}} |
groups with same order and exponent of a group<ul><li>"{{{degree}}}" is not a number.</li> <!--br--><li>"{{{order}}}" is not a number.</li></ul> | groups with same exponent of a group"{{{degree}}}" is not a number. |
cyclic subgroup of order {{{degree}}}.
|
| prime-base logarithm of exponent |
{{{degree p-log}}} |
groups with same order and prime-base logarithm of exponent<ul><li>"{{{degree 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>"{{{degree 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"{{{degree p-log}}}" is not a number. |
|
| nilpotency class |
2 |
groups with same order and nilpotency class"{{{order}}}" is not a number. | groups with same prime-base logarithm of order and nilpotency class"{{{order p-log}}}" is not a number. | groups with same nilpotency class |
|
| derived length |
2 |
groups with same order and derived length"{{{order}}}" is not a number. | groups with same prime-base logarithm of order and derived length"{{{order p-log}}}" is not a number. | groups with same derived length |
the derived subgroup is contained in the cyclic subgroup and is hence abelian
|
| Frattini length |
{{{degree p-log}}} |
groups with same order and Frattini length<ul><li>"{{{degree 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>"{{{degree p-log}}}" is not a number.</li> <!--br--><li>"{{{order p-log}}}" is not a number.</li></ul> | groups with same Frattini length"{{{degree p-log}}}" is not a number. |
|
| minimum size of generating set |
2 |
groups with same order and minimum size of generating set"{{{order}}}" is not a number. | groups with same prime-base logarithm of order and minimum size of generating set"{{{order p-log}}}" is not a number. | groups with same minimum size of generating set |
|
| subgroup rank of a group |
2 |
groups with same order and subgroup rank of a group"{{{order}}}" is not a number. | groups with same prime-base logarithm of order and subgroup rank of a group"{{{order p-log}}}" is not a number. | groups with same subgroup rank of a group |
All proper subgroups are cyclic, dihedral, or Klein four-groups.
|
| rank of a p-group |
2 |
groups with same order and rank of a p-group"{{{order}}}" is not a number. | groups with same prime-base logarithm of order and rank of a p-group"{{{order p-log}}}" is not a number. | groups with same rank of a p-group |
there exist Klein four-subgroups.
|
| normal rank of a p-group |
2 |
groups with same order and normal rank of a p-group"{{{order}}}" is not a number. | groups with same prime-base logarithm of order and normal rank of a p-group"{{{order p-log}}}" is not a number. | groups with same normal rank of a p-group |
all abelian normal subgroups are cyclic.
|
| characteristic rank of a p-group |
2 |
groups with same order and characteristic rank of a p-group"{{{order}}}" is not a number. | groups with same prime-base logarithm of order and characteristic rank of a p-group"{{{order p-log}}}" is not a number. | groups with same characteristic rank of a p-group |
There is a unique (hence characteristic) Klein four-subgroup.
|