Subgroup structure of groups of order 48

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This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 48.
View subgroup structure of group families | View subgroup structure of groups of a particular order |View other specific information about groups of order 48
To understand these in a broader context, see subgroup structure of groups of order 3.2^n | subgroup structure of groups of order 2^4.3^n

Numerical information on counts of subgroups by order

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable 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
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's p^aq^b-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Note that, by Lagrange's theorem, the order of any subgroup must divide the order of the group. Thus, the order of any proper nontrivial subgroup is one of the numbers 2,4,8,16,3,6,12,24.

Here are some observations on the number of subgroups of each order:

  • Congruence condition on number of subgroups of given prime power order: The number of subgroups of order 2 is congruent to 1 mod 2 (i.e., it is odd). The same is true for the number of subgroups of order 4, the number of subgroups of order 8, and the number of subgroups of order 16. Also, the number of subgroups of order 3 is congruent to 1 mod 3.
  • Sylow implies order-conjugate, and hence Sylow number equals index of Sylow normalizer. In particular, it divides the index of the Sylow subgroup. Combining with the congruence condition, we obtain that the number of 2-Sylow subgroups (i.e., subgroups of order 16) is either 1 or 3, and the number of 3-Sylow subgroups (i.e., subgroups of order 3) is either 1, 4, or 16.
  • In the case of a finite nilpotent group, the number of subgroups of each order equals the product of the number of subgroups of order each of its maximal prime power divisors, in the corresponding Sylow subgroup. In particular, we get (number of subgroups of order 3) = 1, (number of subgroups of order 6) = (number of subgroups of order 2), (number of subgroups of order 12) = (number of subgroups of order 4), (number of subgroups of order 24) = (number of subgroups of order 8), and (number of subgroups of order 16) = 1.
  • In the special case of a finite abelian group, we have (number of subgroups of order 3) = (number of subgroups of order 16) = 1, (number of subgroups of order 2) = (number of subgroups of order 8) = (number of subgroups of order 6) = (number of subgroups of order 24), and (number of subgroups of order 4) = (number of subgroups of order 12).