Subgroup structure of Klein four-group
This article gives specific information, namely, subgroup structure, about a particular group, namely: Klein four-group.
View subgroup structure of particular groups | View other specific information about Klein four-group
We use here a Klein four-group with identity element and three non-identity elements all of order two.
We can realize this Klein four-group as , in which case we can set . For more, see element structure of Klein four-group.
Tables for quick information
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (group of prime power order)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so all subgroups have prime power orders)|order of quotient group divides order of group (and equals index of corresponding normal subgroup, so all quotients have prime power orders)
prime power order implies not centerless | prime power order implies nilpotent | prime power order implies center is normality-large
size of conjugacy class of subgroups divides index of center
congruence condition on number of subgroups of given prime power order: The total number of subgroups of any fixed prime power order is congruent to 1 mod the prime.
|Number of subgroups|| 5|
As elementary abelian group of prime-square order for prime :
|Number of conjugacy classes of subgroups||5 (same as number of subgroups, because the group is an abelian group|
|Number of automorphism classes of subgroups|| 3|
As elementary abelian group of order :
|Isomorphism classes of subgroups||trivia group (1 time), cyclic group:Z2 (3 times, all in the same automorphism class), Klein four-group (1 time).|
Table classifying subgroups up to automorphism
|Automorphism class of subgroups||List of subgroups||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||trivial group||1||4||1||1||1||Klein four-group||1||0|
|Z2 in V4||cyclic group:Z2||2||2||3||1||3||cyclic group:Z2||1||1|
|whole group||Klein four-group||4||1||1||1||1||trivial group||0||1|
|Total (3 rows)||--||--||--||--||5||--||5||--||--||--|