# Binary icosahedral group

In

mathematics , the**binary icosahedral group**is an extension of theicosahedral group "I" of order 60 by acyclic group of order 2. It can be defined as thepreimage of the icosahedral group under the 2:1covering homomorphism :$mathrm\{Sp\}(1)\; o\; mathrm\{SO\}(3).,$whereSp(1) is the multiplicative group of unitquaternion s. (For a description of this homomorphism see the article onquaternions and spatial rotation s.) It follows that the binary icosahedral group isdiscrete subgroup of Sp(1) of order 120.It should not be confused with the full icosahedral group, which is a different group of order 120.

**Elements**Explicitly, the binary icosahedral group is given as the union of the 24

Hurwitz unit s:{±1, ±"i", ±"j", ±"k", ½(±1 ± "i" ± "j" ± "k")}with all 96 quaternions obtained from:½(0 ± "i" ± φ^{−1}"j" ± φ"k")by aneven permutation of coordinates (all possible sign combinations). Here φ = ½(1+√5) is thegolden ratio .All told there are 120 elements. They all have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The

convex hull of these 120 elements in 4-dimensional space form aconvex regular 4-polytope called the600-cell .**Properties****Central extension**The binary icosahedral group, denoted by 2"I", is the

universal perfect central extension of the icosahedral group, and thus isquasisimple : it is a perfect central extension of a simple group.Explicitly, it fits into the

short exact sequence :$1\; o\{pm\; 1\}\; o\; 2I\; o\; I\; o\; 1.,$This sequence does not split, meaning that 2"I" is "not" asemidirect product of {±1} by "I". In fact, there is no subgroup of 2"I" isomorphic to "I".The center of 2"I" is the subgroup {±1}, so that the

inner automorphism group is isomorphic to "I". The fullautomorphism group is isomorphic to "S"_{5}(thesymmetric group on 5 letters).The binary icosahedral group is perfect, meaning that it is equal to its

commutator subgroup . In fact, 2"I" is the unique perfect group of order 120. It follows that 2"I" is not solvable.**Isomorphisms**One can show that the binary icosahedral group is isomorphic to the

special linear group SL(2,5) — the group of all 2×2 matrices over thefinite field **F**_{5}with unit determinant; this covers the exceptional isomorphism of $Icong\; A\_5$ with theprojective special linear group PSL(2,5).**Presentation**The group 2"I" has a presentation given by:$langle\; r,s,t\; mid\; r^2\; =\; s^3\; =\; t^5\; =\; rst\; angle$or equivalently,:$langle\; s,t\; mid\; (st)^2\; =\; s^3\; =\; t^5\; angle.$Generators with these relations are given by:$s\; =\; frac\{1\}\{2\}(1+i+j+k)\; qquad\; t\; =\; frac\{1\}\{2\}(varphi+varphi^\{-1\}i+j).$

**ubgroups**The only proper

normal subgroup of 2"I" is the center {±1}.By the

third isomorphism theorem , there is aGalois connection between subgroups of 2"I" and subgroups of "I", where theclosure operator on subgroups of 2"I" is multiplication by {±1}.$-1$ is the only element of order 2, hence it is contained in all subgroups of even order: thus every subgroup of 2"I" is either of odd order or is the preimage of a subgroup of "I".Besides the

cyclic group s generated by the various elements (which can have odd order), the only other subgroups of 2"I" (up to conjugation) are:

*binary dihedral group s of orders 12 and 20 (covering the dihedral groups "D"_{3}and "D"_{5}in "I").

* Thequaternion group consisting of the 8Lipschitz unit s forms a subgroup of index 15, which is also thedicyclic group Dic_{2}; this covers the stabilizer of an edge.

* The 24Hurwitz unit s form an index 5 subgroup called thebinary tetrahedral group ; this covers a chiraltetrahedral group . This group isself-normalizing so itsconjugacy class has 5 members (this gives a map $2I\; o\; S\_5$ whose image is $A\_5$).**Relation to 4-dimensional symmetry groups**The 4-dimensional analog of the icosahedral group is the symmetry group of the

600-cell (also that of the120-cell ). This is theCoxeter group of type "H"_{4}, also denoted [3,3,5] . The rotation subgroup, denoted [3,3,5]^{+}is a group of order 7200 living inSO(4) . SO(4) has a double cover called Spin(4) in much the same way that Sp(1) is the double cover of SO(3). The group Spin(4) is isomorphic to Sp(1)×Sp(1).The preimage of [3,3,5]

^{+}in Spin(4) is precisely the product group 2"I"×2"I" of order 14400. The rotational symmetry group of the 600-cell is then: [3,3,5]^{+}= 2"I"×2"I"/{±1}.Various other 4-dimensional symmetry groups can be constructed from 2"I". For details, see (Conway and Smith, 2003).

**Applications**The

coset space Sp(1)/2"I" is aspherical 3-manifold called thePoincaré homology sphere . It is an example of ahomology sphere , i.e. a 3-manifold whosehomology group s are identical to those of a3-sphere . Thefundamental group of the Poincaré sphere is isomorphic to the binary icosahedral group.**ee also***

binary polyhedral group

*binary cyclic group

*binary dihedral group

*binary tetrahedral group

*binary octahedral group **References***cite book | first = John H. | last = Conway | coauthors = Smith, Derek A. | authorlink = John Horton Conway | title = On Quaternions and Octonions | publisher = AK Peters, Ltd | location = Natick, Massachusetts | year = 2003 | isbn = 1-56881-134-9

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