Subgroup structure of alternating group:A5
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This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A5.
View subgroup structure of particular groups | View other specific information about alternating group:A5
This article discusses the subgroup structure of alternating group:A5, which is the alternating group on the set . The group has order 60.
Family contexts
Family name | Parameter values | General discussion of subgroup structure of family |
---|---|---|
alternating group | degree , i.e., the group | subgroup structure of alternating groups |
projective special linear group of degree two | field:F5, i.e., it is the group | subgroup structure of projective special linear group of degree two over a finite field |
projective special linear group of degree two | field:F4, i.e., it is the group | subgroup structure of projective special linear group of degree two over a finite field |
Tables for quick information
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate
Quick summary
Item | Value |
---|---|
number of subgroups | 59 Compared with : 2, 10, 59, 501, 3786, 48337, ... |
number of conjugacy classes of subgroups | 9 Compared with : 2, 5, 9, 22, 40, 137, ... |
number of automorphism classes of subgroups | 9 Compared with : 2, 5, 9, 16, 37, 112, ... |
isomorphism classes of Sylow subgroups and the corresponding fusion systems | 2-Sylow: Klein four-group (order 4) as V4 in A5 (with its simple fusion system -- see simple fusion system for Klein four-group). Sylow number is 5. 3-Sylow: cyclic group:Z3 (order 3) as Z3 in A5. Sylow number is 10. 5-Sylow: cyclic group:Z5 (order 5) as Z5 in A5. Sylow number is 6. |
Hall subgroups | In addition to the whole group, trivial subgroup, and Sylow subgroups: -Hall subgroup of order 12 (A4 in A5). There is no -Hall subgroup or -Hall subgroup. |
maximal subgroups | maximal subgroups have orders 6 (twisted S3 in A5), 10 (D10 in A5), 12 (A4 in A5) |
normal subgroups | only the whole group and the trivial subgroup, because the group is simple. See alternating groups are simple. |
Table classifying subgroups up to automorphisms
Note that A5 is simple, and hence no proper nontrivial subgroup is normal or subnormal.
Automorphism class of subgroups | Representative subgroup (full list if small, generating set if large) | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes | Size of each conjugacy class | Total number of subgroups | Note |
---|---|---|---|---|---|---|---|---|
trivial subgroup | trivial group | 1 | 60 | 1 | 1 | 1 | trivial | |
subgroup generated by double transposition in A5 | cyclic group:Z2 | 2 | 30 | 1 | 15 | 15 | ||
V4 in A5 | Klein four-group | 4 | 15 | 1 | 5 | 5 | 2-Sylow | |
A3 in A5 | cyclic group:Z3 | 3 | 20 | 1 | 10 | 10 | 3-Sylow | |
twisted S3 in A5 | symmetric group:S3 | 6 | 10 | 1 | 10 | 10 | maximal | |
A4 in A5 | alternating group:A4 | 12 | 5 | 1 | 5 | 5 | 2,3-Hall, maximal | |
Z5 in A5 | cyclic group:Z5 | 5 | 12 | 1 | 6 | 6 | 5-Sylow | |
D10 in A5 | dihedral group:D10 | 10 | 6 | 1 | 6 | 6 | maximal | |
whole group | alternating group:A5 | 60 | 1 | 1 | 1 | 1 | ||
Total | -- | -- | -- | -- | 9 | -- | 59 | -- |
Table classifying isomorphism types of subgroups
Group name | Order | Second part of GAP ID (first part is order) | Occurrences as subgroup | Conjugacy classes of occurrence as subgroup | Automorphism classes of occurrence as subgroup | Occurrences as normal subgroup | Occurrences as characteristic subgroup |
---|---|---|---|---|---|---|---|
trivial group | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
cyclic group:Z2 | 2 | 1 | 15 | 1 | 1 | 0 | 0 |
cyclic group:Z3 | 3 | 1 | 10 | 1 | 1 | 0 | 0 |
Klein four-group | 4 | 2 | 5 | 1 | 1 | 0 | 0 |
cyclic group:Z5 | 5 | 1 | 6 | 1 | 1 | 0 | 0 |
symmetric group:S3 | 6 | 1 | 10 | 1 | 1 | 0 | 0 |
dihedral group:D10 | 10 | 1 | 6 | 1 | 1 | 0 | 0 |
alternating group:A4 | 12 | 3 | 5 | 1 | 1 | 0 | 0 |
alternating group:A5 | 60 | 5 | 1 | 1 | 1 | 1 | 1 |
Total | -- | -- | 59 | 9 | 9 | 2 | 2 |