Subgroup structure of cyclic group:Z4

From Groupprops
Jump to: navigation, search
This article gives specific information, namely, subgroup structure, about a particular group, namely: cyclic group:Z4.
View subgroup structure of particular groups | View other specific information about cyclic group:Z4
Z4latticeofsubgroups.png

Table classifying subgroups up to automorphism


Note that since the group is a cyclic group, all subgroups are characteristic subgroups (see cyclic implies every subgroup is characteristic) and in particular normal subgroups.

Automorphism class of subgroups List of subgroups (power notation, generator a) List of subgroups (\mathbb{Z}/4\mathbb{Z} notation) Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes(=1 iff automorph-conjugate subgroup) Size of each conjugacy class(=1 iff normal subgroup) Total number of subgroups(=1 iff characteristic subgroup) Isomorphism class of quotient (if exists) Subnormal depth Nilpotency class
trivial subgroup \{ e \} \{ 0 \} trivial group 1 4 1 1 1 cyclic group:Z4 1 0
Z2 in Z4 \{ e, a^2 \} \{ 0, 2 \} cyclic group:Z2 2 2 1 1 1 cyclic group:Z2 1 1
whole group \{ e, a, a^2, a^3 \} \{ 0,1,2,3 \} cyclic group:Z4 4 1 1 1 1 trivial group 1 1
Total (3 rows) -- -- -- -- -- 3 -- 3 -- -- --