1.
Icosahedral symmetry
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A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as A5, the latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation, and Coxeter diagram. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are, I, ⟨ s, t ∣ s 2, t 3,5 ⟩ I h, ⟨ s, t ∣ s 3 −2, t 5 −2 ⟩ and these correspond to the icosahedral groups being the triangle groups. The first presentation was given by William Rowan Hamilton in 1856, note that other presentations are possible, for instance as an alternating group. The icosahedral rotation group I is of order 60, the group I is isomorphic to A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the group contains 5 versions of Th with 20 versions of D3, and 6 versions of D5. The full icosahedral group Ih has order 120 and it has I as normal subgroup of index 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the corresponding to element. Ih acts on the compound of five cubes and the compound of five octahedra and it acts on the compound of ten tetrahedra, I acts on the two chiral halves, and −1 interchanges the two halves. Notably, it does not act as S5, and these groups are not isomorphic, the group contains 10 versions of D3d and 6 versions of D5d. I is also isomorphic to PSL2, but Ih is not isomorphic to SL2, all of these classes of subgroups are conjugate, and admit geometric interpretations. Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate. Stabilizers of a pair of edges in Ih give Z2 × Z2 × Z2, there are 5 of these, stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate. g. Flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, in aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011. Icosahedral symmetry is equivalently the projective linear group PSL, and is the symmetry group of the modular curve X. The modular curve X is geometrically a dodecahedron with a cusp at the center of each polygonal face, similar geometries occur for PSL and more general groups for other modular curves

2.
Dihedral group of order 6
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In mathematics, the smallest non-abelian group has 6 elements. It is a group with notation D3 and the symmetric group of degree 3. This page illustrates many group concepts using this group as example, in two dimensions, the group D3 is the symmetry group of an equilateral triangle. In contrast with the case of a square or other polygon, all permutations of the vertices can be achieved by rotation, let a be the action swap the first block and the second block, and let b be the action swap the second block and the third block. In multiplicative form, we traditionally write xy for the combined action first do y, then do x, so that ab is the action RGB ↦ RBG ↦ BRG, i. e. take the last block and move it to the front. Note that the action aa has the effect RGB ↦ GRB ↦ RGB, leaving the blocks as they were, similarly, bb = e, = e, and = = e, so each of the above actions has an inverse. By inspection, we can determine associativity and closure, note for example that a = a = aba. The group is non-abelian since, for example, ab ≠ ba, since it is built up from the basic actions a and b, we say that the set generates it. Note that the second means that the group is a Coxeter group. With the generators a and b, we define the additional shorthands c, = aba, d, = ab, in the form of a Cayley table, the group operations now read, Note that non-equal non-identity elements only commute if they are each others inverse. Therefore the group is centerless, i. e. the center of the group consists only of the identity element, therefore, if we apply, then, and then the inverse of, which is also, the resulting permutation is. Note that conjugate group elements always have the order. From Lagranges theorem we know that any subgroup of a group with 6 elements must have order 2 or 3. The existence of subgroups of order 2 and 3 is also a consequence of Cauchys theorem, the first-mentioned is, the alternating group A3. The left cosets and the cosets of A3 coincide and consist of A3. The left cosets of are, The right cosets of are, Thus A3 is normal, the quotient group G / A3 is isomorphic with C2. G = A3 ⋊ H, a product, where H is a subgroup of two elements, and one of the three swaps. This decomposition is also a consequence of the Schur–Zassenhaus theorem, in terms of permutations the two group elements of G / A3 are the set of even permutations and the set of odd permutations

3.
Alternating group
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In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the group of degree n, or the alternating group on n letters. For n >1, the group An is the subgroup of the symmetric group Sn with index 2 and has therefore n. It is the kernel of the signature group homomorphism sgn. The group An is abelian if and only if n ≤3 and simple if, a5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group. The group A4 has a Klein four-group V as a normal subgroup, namely the identity and the double transpositions. As in the group, the conjugacy classes in An consist of elements with the same cycle shape. Examples, The two permutations and are not conjugates in A3, although they have the same cycle shape, the permutation is not conjugate to its inverse in A8, although the two permutations have the same cycle shape, so they are conjugate in S8. An is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions and this generating set is often used to prove that An is simple for n ≥5. For n =1 and 2, the group is trivial. For n =3 the automorphism group is Z2, with trivial inner automorphism group, the outer automorphism group of A6 is the Klein four-group V = Z2 × Z2, and is related to the outer automorphism of S6. The extra outer automorphism in A6 swaps the 3-cycles with elements of shape 32, there are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are, A4 is isomorphic to PSL2 and the group of chiral tetrahedral symmetry. A5 is isomorphic to PSL2, PSL2, and the group of chiral icosahedral symmetry. A6 is isomorphic to PSL2 and PSp4, more obviously, A3 is isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group. A subgroup of three elements with any additional element generates the whole group, for all n ≠4, An has no nontrivial normal subgroups. Thus, An is a group for all n ≠4. A5 is the smallest non-solvable group, the group homology of the alternating groups exhibits stabilization, as in stable homotopy theory, for sufficiently large n, it is constant. However, there are some low-dimensional exceptional homology, note that the homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions

4.
Klein four-group
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In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe by Felix Klein in 1884, with four elements, the Klein four-group is the smallest non-cyclic group, and the cyclic group of order 4 and the Klein four-group are, up to isomorphism, the only groups of order 4. The smallest non-abelian group is the group of degree 3. The Klein groups Cayley table is given by, The Klein four-group is also defined by the group presentation V = ⟨ a, b ∣ a 2 = b 2 =2 = e ⟩. All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation, the Klein four-group is the smallest non-cyclic group. It is however a group, and isomorphic to the dihedral group of order 4, Dih2, other than the group of order 2. The Klein four-group is also isomorphic to the direct sum Z2 ⊕ Z2, so that it can be represented as the pairs under component-wise addition modulo 2, the Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. Another numerical construction of the Klein four-group is the set, with the operation being multiplication modulo 8, here a is 3, b is 5, and c = ab is 3 ×5 =15 ≡7. The three elements of two in the Klein four-group are interchangeable, the automorphism group of V is the group of permutations of these three elements. In fact, it is the kernel of a group homomorphism from S4 to S3. In the construction of finite rings, eight of the rings with four elements have the Klein four-group as their additive substructure. The quotient group / is isomorphic to the Klein four-group, in a similar fashion, the group of units of the split-complex number ring, when divided by its identity component, also results in the Klein four-group. The Klein four-group as a subgroup of the alternating group A4 is not the group of any simple graph. It is, however, the group of a two-vertex graph where the vertices are connected to each other with two edges, making the graph non-simple. A. Armstrong Groups and Symmetry, Springer Verlag, page 53, W. E. Barnes Introduction to Abstract Algebra, D. C

5.
Finite group
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In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, with a finite group, the set is finite. As a consequence, the classification of finite simple groups was achieved. During the second half of the century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups. One such family of groups is the family of linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, the theory of Lie groups, which may be viewed as dealing with continuous symmetry, is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space, the properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry. Since there are n. possible permutations of a set of n symbols, a cyclic group Zn is a group all of whose elements are powers of a particular element a where an = a0 = e, the identity. A typical realization of this group is as the nth roots of unity. Sending a to a root of unity gives an isomorphism between the two. This can be done with any finite cyclic group, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order. They are named after Niels Henrik Abel, an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of an abelian group can be described directly in terms of these invariants. A group of Lie type is a closely related to the group G of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type give the bulk of nonabelian simple groups. Special cases include the groups, the Chevalley groups, the Steinberg groups. The systematic exploration of finite groups of Lie type started with Camille Jordans theorem that the special linear group PSL is simple for q ≠2,3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL of finite simple groups, other classical groups were studied by Leonard Dickson in the beginning of 20th century

6.
Dicyclic group
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In group theory, a dicyclic group is a member of a class of non-abelian groups of order 4n. It is an extension of the group of order 2 by a cyclic group of order 2n. In the notation of exact sequences of groups, this extension can be expressed as,1 → C2 n → Dic n → C2 →1, more generally, given any finite abelian group with an order-2 element, one can define a dicyclic group. Some things to note which follow from this definition, x4 =1 x2ak = ak+n = akx2 if j = ±1, thus, every element of Dicn can be uniquely written as akxj, where 0 ≤ k < 2n and j =0 or 1. When n =2, the group is isomorphic to the quaternion group Q. More generally, when n is a power of 2, the group is isomorphic to the generalized quaternion group. For each n >1, the dicyclic group Dicn is a group of order 4n. Let A = <a> be the subgroup of Dicn generated by a, then A is a cyclic group of order 2n, so =2. As a subgroup of index 2 it is automatically a normal subgroup, the quotient group Dicn/A is a cyclic group of order 2. Dicn is solvable, note that A is normal, and being abelian, is itself solvable, there is a superficial resemblance between the dicyclic groups and dihedral groups, both are a sort of mirroring of an underlying cyclic group. But the presentation of a group would have x2 =1, instead of x2 = an. In particular, Dicn is not a product of A and <x>. The dicyclic group has an involution, namely x2 = an. Note that this element lies in the center of Dicn, indeed, the center consists solely of the identity element and x2. If we add the relation x2 =1 to the presentation of Dicn one obtains a presentation of the dihedral group Dih2n, there is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations. Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism, the answer is just the dihedral symmetry group Dihn. For this reason the group is also known as the binary dihedral group. Note that the group does not contain any subgroup isomorphic to Dihn

7.
Frobenius group
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In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius, a subgroup H of a Frobenius group G fixing a point of the set X is called the Frobenius complement. The identity element together with all elements not in any conjugate of H form a subgroup called the Frobenius kernel K. The Frobenius group G is the product of K and H, G = K ⋊ H. Both the Frobenius kernel and the Frobenius complement have very restricted structures, J. G. Thompson proved that the Frobenius kernel K is a nilpotent group. If H has even order then K is abelian, the Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic, this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called a Z-group, and in particular must be a metacyclic group, this means it is the extension of two cyclic groups. If a Frobenius complement H is not solvable then Zassenhaus showed that it has a subgroup of index 1 or 2 that is the product of SL2. In particular, if a Frobenius complement coincides with its derived subgroup, if a Frobenius complement H is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points. The Frobenius kernel K is uniquely determined by G as it is the Fitting subgroup, in particular a finite group G is a Frobenius group in at most one way. The smallest example is the group on 3 points, with 6 elements. The Frobenius kernel K has order 3, and the complement H has order 2, for every finite field Fq with q elements, the group of invertible affine transformations x ↦ a x + b, a ≠0 acting naturally on Fq is a Frobenius group. The preceding example corresponds to the case F3, the field with three elements, identifying F8* with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ=x² of F8 and τ to be multiplication by any element not in the prime field F2. This Frobenius group acts transitively on the 21 flags in the Fano plane. The dihedral group of order 2n with n odd is a Frobenius group with complement of order 2. More generally if K is any group of odd order and H has order 2 and acts on K by inversion. Many further examples can be generated by the following constructions, if we replace the Frobenius complement of a Frobenius group by a non-trivial subgroup we get another Frobenius group. If we have two Frobenius groups K1. H and K2. H then. H is also a Frobenius group

8.
Lagrange's theorem (group theory)
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Lagranges theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange and this can be shown using the concept of left cosets of H in G. If we can show that all cosets of H have the number of elements. This map is bijective because its inverse is given by f −1 = a b −1 y and this proof also shows that the quotient of the orders |G| / |H| is equal to the index. If we allow G and H to be infinite, and write this statement as | G | = ⋅ | H |, then, seen as a statement about cardinal numbers, it is equivalent to the axiom of choice. A consequence of the theorem is that the order of any element a of a group divides the order of that group. If the group has n elements, it follows a n = e and this can be used to prove Fermats little theorem and its generalization, Eulers theorem. These special cases were known long before the general theorem was proved, the theorem also shows that any group of prime order is cyclic and simple. This in turn can be used to prove Wilsons theorem, that if p is prime p is a factor of. Hence p < q, contradicting the assumption that p is the largest prime, Lagranges theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. This does not hold in general, given a finite group G, the smallest example is the alternating group G = A4, which has 12 elements but no subgroup of order 6. A CLT group is a group with the property that for every divisor of the order of the group. It is known that a CLT group must be solvable and that every group is a CLT group. There are partial converses to Lagranges theorem, for solvable groups, Halls theorems assert the existence of a subgroup of order equal to any unitary divisor of the group order. Lagrange did not prove Lagranges theorem in its general form, the number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations that preserve the polynomial. So the size of H divides n, with the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name. In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagranges theorem for the case of Z*, the multiplicative group of nonzero integers modulo p. In 1844, Augustin-Louis Cauchy proved Lagranges theorem for the symmetric group Sn, camille Jordan finally proved Lagranges theorem for the case of any permutation group in 1861