Template:Abelian p-group arithmetic function table

Want to compare and contrast arithmetic function values with other groups of the same order? Check out [[groups of order {{{order}}}#Arithmetic functions]]

Function	Value	Similar groups	Explanation for function value
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