Sylow theory in locally finite groups

Abstract

In this thesis we study extensions of the classical Sylow theory from finite groups to the class of locally finite groups. To this end we present theorems from P. Hall as an introduction to π-groups in finite groups and theorems from A. O. Asar and B. Hartley to introduce Sylow theory in locally finite groups. We then conclude the thesis with touching on more well-behaved groups (for example soluble groups) and where conjugacy holds between Sylow subgroups as well as briefly going over why this does not hold in general.

Popular Abstract

Say we have a set of arbitrary elements (the mathematical term for objects or things in a set). Then a group is a set with an operation (for example addition + or multiplication ×) that combines any two elements of the set to produce a third element within the same set and a few additional conditions. Taking a few of the elements from our group we get a subset. If our subset also forms a group under the same operation as the group we took it from it is called a subgroup. An example of a group would be an analog clock. If you add any two hours on a clock the resulting time is still representable by the same clock. A subgroup could then be the hours 3, 6, 9 and 12 as adding together any of these numbers we will never reach the other positions on the clock.
This is but one example of where group theory enters our everyday life. Its origins are from the middle of the 19th century and it has ever since been used widely in mathematics and other subjects where the need for study of symmetry is important, such as classifying particles in physics and understanding molecular structures in chemistry.
We call the number of elements in a group or subgroup its order. If the order of a subgroup is of a prime number to the largest power (for example 2^2 in 12 = 2^2 · 3) so that it still divides the order of the group, we get a Sylow subgroup. If we take our clock example which has order 12 = 3 × 2^2, we have Sylow subgroups of order 3 and 2^2, but the subgroups of order 2 are not Sylow subgroups.
In this thesis, we study the difference between Sylow subgroups in finite and locally finite groups. An example of a finite group would be a clock with n hours. We can choose n to be any finite number, where we usually choose 12 or 24. This group is finite as we can only move the clock to finitely many positions (we only use whole hours). If the clock had an infinite amount of hours, then the digit would never make a full turn, so there are an infinite number
of positions to which it can move. A locally finite group would be a 12-hour clock with infinite clocks nesting inside it. We are familiar with minutes and seconds. However, if we continue with an infinite number of divisions, then it has an infinite number of positions it can move to, yet for all subgroups (only hours, minutes, seconds, etc) we have a finite number of positions.
When working with only finite groups, we can work with their order and can get structures in Sylow subgroups such as conjugacy. Conjugacy helps classify elements of a group into conjugacy classes, which reveal important structural properties of the group such as symmetries, where as an example we take the clock again. This is a group Z12 (integers 0, 1, . . . , 11) and has 12 conjugacy classes, one for each hour. If we have a transparent clock and allow flipping
it to the back side, so that the numbers run backwards, we instead get the dihedral group D12 of order 24 (there are 12 hours and 12 flips, one for each hour). Now, the clock has more symmetries and those symmetries can move one element into another. Different rotations become equivalent, and many reflections become equivalent. More symmetries mean more elements get grouped together, so conjugacy classes become larger. An example of this is when we start at 12 and flip the clock, then rotating forwards by one hour on a reversed clock gets us to 11, if we now flip again to un-reverse the clock we get the same result as if we just rotated backwards once.
As there can be infinitely many elements in a locally finite group the order is then infinite. This then breaks our definition of Sylow subgroups as we can no longer tell what primes divide the order of our group. By changing the definition, we can still work with Sylow subgroups. However, the theorems need to be reworked and many will simply fail when the order of our group is no longer finite. The conjugacy of the Sylow subgroups becomes contingent on many things and is no longer guaranteed. In general, there are many infinite groups that behave badly in the context of Sylow theory.

• author: Hentschel, Ansgar
• supervisor: Anitha Thillaisundaram
• organization: Mathematics (Faculty of Engineering); Mathematics (Faculty of Sciences); Centre for Mathematical Sciences
• course: MATK11 20252
• year: 2026
• type: M2 - Bachelor Degree
• subject: Mathematics and Statistics
• keywords: Locally finite, Sylow theory, Hall theory, infinite group theory, A.O.Asar and B.Hartley theorems
• publication/series: Bachelor’s Theses in Mathematical Sciences
• report number: LUNFMA-4186-2026
• ISSN: 1654-6229
• other publication id: 2026:K2
• language: English
• id: 9225311
• date added to LUP: 2026-05-18 13:59:20
• date last changed: 2026-05-18 13:59:20