Page 93 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 93
mož Moravec: Some Topics in the Theory of Finite Groups 81
20. Show that a group of order 40 cannot be simple. Do the same for groups of order
84.
21. Prove that Sn is given by a presentation listed in Example 3.2.8.
22. Show that A4 has a presentation 〈x , y | x 2 = y 3 = (x y )3 = 1〉.
23. Identify the group 〈x , y , z | z y = z 2, x z = x 2, y x = y 2〉.
24. Find all the composition series of S4.
3.3 Finite simple groups
Quote from Wikipedia:
In mathematics, the classification of finite simple groups states that every
finite simple group is cyclic, or alternating, or in one of 16 families of groups
of Lie type, or one of 26 sporadic groups... These groups can be seen as the
basic building blocks of all finite groups, in a way reminiscent of the way the
prime numbers are the basic building blocks of the natural numbers. The
Jordan-Hoölder theorem is a more precise way of stating this fact about fi-
nite groups. However, a significant difference with respect to the case of inte-
ger factorization is that such “building blocks” do not necessarily determine
uniquely a group, since there might be many non-isomorphic groups with
the same composition series or, put in another way, the extension problem
does not have a unique solution.
The proof of the theorem consists of tens of thousands of pages in several
hundred journal articles written by about 100 authors, published mostly be-
tween 1955 and 2004. Gorenstein (d.1992), Lyons, and Solomon are gradu-
ally publishing a simplified and revised version of the proof.
3.3.1 Faithful primitive actions and Iwasawa’s Lemma
In this section we prove Iwasawa’s Lemma which provides a useful criterion for simplicity
of a given finite group.
Transitive actions
Let H be a subgroup of G . Denote by H \G the set of right cosets of H in G (note that,
unless H is a normal subgroup, H \G is only a set, not a group in general). The group G
acts on H \G by right multiplication. This action is obviously transitive. Our first result
shows that this example is, in a sense, generic. Before stating this in a precise form, we
need a definition. Let G act on sets X1 and X2. An equivalence between these two actions
is a bijection f : X1 → X2 such that (x g )f = (x f )g for all x ∈ X1 and g ∈ G .
20. Show that a group of order 40 cannot be simple. Do the same for groups of order
84.
21. Prove that Sn is given by a presentation listed in Example 3.2.8.
22. Show that A4 has a presentation 〈x , y | x 2 = y 3 = (x y )3 = 1〉.
23. Identify the group 〈x , y , z | z y = z 2, x z = x 2, y x = y 2〉.
24. Find all the composition series of S4.
3.3 Finite simple groups
Quote from Wikipedia:
In mathematics, the classification of finite simple groups states that every
finite simple group is cyclic, or alternating, or in one of 16 families of groups
of Lie type, or one of 26 sporadic groups... These groups can be seen as the
basic building blocks of all finite groups, in a way reminiscent of the way the
prime numbers are the basic building blocks of the natural numbers. The
Jordan-Hoölder theorem is a more precise way of stating this fact about fi-
nite groups. However, a significant difference with respect to the case of inte-
ger factorization is that such “building blocks” do not necessarily determine
uniquely a group, since there might be many non-isomorphic groups with
the same composition series or, put in another way, the extension problem
does not have a unique solution.
The proof of the theorem consists of tens of thousands of pages in several
hundred journal articles written by about 100 authors, published mostly be-
tween 1955 and 2004. Gorenstein (d.1992), Lyons, and Solomon are gradu-
ally publishing a simplified and revised version of the proof.
3.3.1 Faithful primitive actions and Iwasawa’s Lemma
In this section we prove Iwasawa’s Lemma which provides a useful criterion for simplicity
of a given finite group.
Transitive actions
Let H be a subgroup of G . Denote by H \G the set of right cosets of H in G (note that,
unless H is a normal subgroup, H \G is only a set, not a group in general). The group G
acts on H \G by right multiplication. This action is obviously transitive. Our first result
shows that this example is, in a sense, generic. Before stating this in a precise form, we
need a definition. Let G act on sets X1 and X2. An equivalence between these two actions
is a bijection f : X1 → X2 such that (x g )f = (x f )g for all x ∈ X1 and g ∈ G .
