Subgroup structure of Mathieu group:M10: Difference between revisions
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group = Mathieu group:M10| | group = Mathieu group:M10| | ||
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==Tables for quick information== | |||
{{finite group subgroup structure facts to check against}} | |||
<section begin="summary"/> | |||
===Quick summary=== | |||
{| class="sortable" border="1" | |||
! Item !! Value | |||
|- | |||
| [[number of subgroups]] || 793 | |||
|- | |||
| [[number of conjugacy classes of subgroups]] || 25 | |||
|- | |||
| [[number of automorphism classes of subgroups]] || 24 | |||
|- | |||
| isomorphism classes of [[Sylow subgroup]]s and the corresponding [[Sylow number]]s || 2-Sylow: [[semidihedral group:SD16]], Sylow number is 45<br>3-Sylow: [[elementary abelian group:E9]], Sylow number is 10<br>5-Sylow: [[cyclic group:Z5]], Sylow number is 36 | |||
|- | |||
| [[Hall subgroup]]s || No Hall subgroups other than the trivial subgroup, whole group, and Sylow subgroups | |||
|- | |||
| [[maximal subgroup]]s || maximal subgroups have order 16, 20, 72, 360 | |||
|- | |||
| [[normal subgroup]]s || the only proper nontrivial normal subgroup is of index two and is isomorphic to [[alternating group:A6]] (order 360) | |||
|} | |||
<section end="summary"/> | |||
Latest revision as of 01:36, 1 June 2012
This article gives specific information, namely, subgroup structure, about a particular group, namely: Mathieu group:M10.
View subgroup structure of particular groups | View other specific information about Mathieu group:M10
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 | 793 |
| number of conjugacy classes of subgroups | 25 |
| number of automorphism classes of subgroups | 24 |
| isomorphism classes of Sylow subgroups and the corresponding Sylow numbers | 2-Sylow: semidihedral group:SD16, Sylow number is 45 3-Sylow: elementary abelian group:E9, Sylow number is 10 5-Sylow: cyclic group:Z5, Sylow number is 36 |
| Hall subgroups | No Hall subgroups other than the trivial subgroup, whole group, and Sylow subgroups |
| maximal subgroups | maximal subgroups have order 16, 20, 72, 360 |
| normal subgroups | the only proper nontrivial normal subgroup is of index two and is isomorphic to alternating group:A6 (order 360) |