Cyclic group:Z49

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Definition

This group, denoted $C_{49},\mathbb {Z} _{49}$ or $\mathbb {Z} /49\mathbb {Z}$, is defined in the following equivalent ways:

1. It is a cyclic group of order 49.

Relationships to other groups

It is one of two groups of order 49, since 49 is the square of the prime number 7. The other group is elementary abelian group:E49.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 49#Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	49	groups with same order
exponent of a group	49	groups with same order and exponent of a group | groups with same exponent of a group
nilpotency class	1	groups with same order and nilpotency class | groups with same nilpotency class	cyclic implies abelian
derived length	1	groups with same order and derived length | groups with same derived length	cyclic implies abelian

GAP implementation

Group ID

This finite group has order 49 and has ID 1 among the groups of order 49 in GAP's SmallGroup library. For context, there are groups of order 49. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(49,1)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(49,1);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [49,1]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

Description	Functions used
CyclicGroup(49)	CyclicGroup