Template:Abelian p-group arithmetic function table

From Groupprops

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
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