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Concept
Covering groups of the alternating and symmetric groups
Summary
In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classified in : for n ≥ 4, the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold.
For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4.
A group homomorphism from D to G is said to be a Schur cover of the finite group G if:
the kernel is contained both in the center and the commutator subgroup of D, and
amongst all such homomorphisms, this D has maximal size.
The Schur multiplier of G is the kernel of any Schur cover and has many interpretations. When the homomorphism is understood, the group D is often called the Schur cover or Darstellungsgruppe.
The Schur covers of the symmetric and alternating groups were classified in . The symmetric group of degree n ≥ 4 has
Schur covers of order 2⋅n! There are two isomorphism classes if n ≠ 6 and one isomorphism class if n = 6.
The alternating group of degree n has one isomorphism class of Schur cover, which has order n! except when n is 6 or 7, in which case the Schur cover has order 3⋅n!.
Schur covers can be described by means of generators and relations. The symmetric group Sn has a presentation on n−1 generators ti for i = 1, 2, ..., n−1 and relations
titi = 1, for 1 ≤ i ≤ n−1
ti+1titi+1 = titi+1ti, for 1 ≤ i ≤ n−2
tjti = titj, for 1 ≤ i < i+2 ≤ j ≤ n−1.
These relations can be used to describe two non-isomorphic covers of the symmetric group. One covering group has generators z, t1, ..., tn−1 and relations:
zz = 1
titi = z, for 1 ≤ i ≤ n−1
ti+1titi+1 = titi+1ti, for 1 ≤ i ≤ n−2
tjti = titjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
Official source
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