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Hexagon Symmetry Groups
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0:02
Lets look at the symmetries of a hexagon. There are six rotational symmetries, one of which leaves the hexagon unchanged. Additionally, there are six reflectional symmetries. An important property of symmetries is that combining two of them always results in another symmetry. For example, first rotating by 60 degrees and then reflecting the hexagon, results in the same symmetry as reflecting the hexagon along a different axis. This means we have an operation that combines two
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symmetries into another symmetry. A structure with such an operation is called a group. We define a group to be a set with an operation, which satisfies four properties. The first property is Closure. We already saw, that when composing two symmetries, you get another symmetry. More generally, combining any two elements in the group results in another element of the group. The next property is associativity. Here we require that the way elements are grouped in the
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composition does not matter, so the parentheses can be arranged without changing the outcome. Then we have the existence of an identity element. For instance, the zero degree rotation does nothing, so we can compose this with any symmetry and it does not change the symmetry. In general, there should exists an element that, when combined with any other element, leaves it unchanged. The final property of a group is the existence of inverse elements. For every symmetry,
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there is another symmetry which reverses the first one. For every element in the group, there is another element, such that their composition is the identity element. Using these four properties, we can derive many useful results about groups. For example, the identity element of a group is always unique. To prove this, assume there are two identity elements. Then we compose them. By using that both elements are identity elements, we get that the product is equal to
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both identity elements. This implies that the identity element of a group is unique. We can also easily prove that the inverse of an element is unique. We denote the inverse of an element using an exponent of -1. We often drop the operation symbol when writing compositions. Let's look at some examples of groups. We already saw, that the symmetries of a hexagon form a group. This is called the dihedral group
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D6. The identity element is the rotation of zero degrees, The inverse of the rotation by 120 degrees is the rotation by minus 120 degrees. Every reflection is its own inverse. More generally, the dihedral group Dn describes the symmetries of a regular n-gon. It consists of n rotations and n reflections, giving it a total of 2n elements. On the natural numbers, we naturally have addition as an operation. This operation
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is associative and zero is the identity element, but we don't have inverses, as negative numbers are not part of the natural numbers. Instead we can look at the integers with addition, which form a group. This group does have the additional property of being abelian, which means that we can interchange the two elements, we are applying the operation to, without changing the value. Another example of a group is the symmetric group. Here the elements are permutations of n elements.
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For instance this is a permutation of 9 elements. We can always decompose a permutation in cycles. Another operation, which allows us to form a group, is matrix multiplication. Here the set of invertible n times n matrices with entries in some field form a group. The identity element is the matrix with ones on the diagonal and zeros elsewhere. When studying groups, its often convenient to study subsets that also form groups under the
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same operation. Such subsets are called subgroups. A subset forms a subgroup if and only if the identity element is in the subset and the subset is closed under taking products and inverses. The smallest subgroup is always the subset with just the identity element. In the integers, the set of even integers form a subgroup, as the sum and difference of even integers is still even. More generally for any integer n, all the multiples of n form a subgroup of the integers.
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To find more subgroups, we define powers of an element. For an element g, we define g to the power of n as g composed with itself n times. The zeroth power is the identity, and negative powers correspond to the inverse. The smallest subgroup containing g consists of all its powers. For example, the subgroup generated by a 60-degree rotation contains all rotations of the hexagon, while the subgroup generated by a 120-degree rotation consists of just three rotations.
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An important subgroup is the center of a group. This consists of all elements that commute with every element in the group. A group is abelian if and only if its center is the entire group. For example, the center of dihedral group D6 consists of just the identity element and the 180-degree rotation. An important result to understand the subgroups of finite groups is Lagranges theorem, which states
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that the number of elements in a subgroup divides the number of elements of the entire group. To illustrate this, consider a group as a set of points. A subgroup is a subset of these points. The rest of the elements can be grouped into sets of equal size, called cosets. The key fact is that cosets partition the entire group into equal-sized pieces. The total number of elements in the group
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is the product of the number of cosets and the size of the subgroup, proving Lagrange’s Theorem. For example, in the dihedral group D6, which has 12 elements, possible subgroup sizes must divide 12. The subgroup of all rotations has 6 elements, and a subgroup consisting of the identity and a single reflection has 2 elements. If a group has a prime number p of elements, the only possible subgroups have sizes 1 and p, so only the identity element or the entire gorup.
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This means that any element other than the identity generates the entire group, making it a cyclic group. A cyclic group is a group generated by a single element. A natural question to ask is, whether the converse of Lagrange's theorem also holds. So if some number divides the order of a group, is there a subgroup with that order? Generally this is not the case. However, for example when we are just looking at prime powers, this statement is true.
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To understand groups better, we need to know, how different groups relate to each other. So suppose we have two groups G and H. We can look at maps from G to H, which preserve the group structure, meaning the map f translates the operation of G into the operation of H. Such a map is called a group homomorphism. Lets look at a few examples. We know that the determinant is multiplicative, meaning it turns matrix multiplication into multiplication of numbers. . This
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makes it a group homomorphism from the general linear group of matrices to the multiplicative group of a field. Another example is the exponentiation map, which converts addition into multiplication. Specifically, it defines a homomorphism from the complex numbers under addition to the nonzero complex numbers under multiplication. For any group, there is always a homomorphism from the integers into that group, which maps each integer to the corresponding power of a given element.
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A bijective homomorphism is called an isomorphism. A function is bijective if every element in the target group corresponds to exactly one element from the original group. If there is an isomorphism between two groups, we say that they are isomorphic and denote it like this. Isomorphic groups have the same number of elements and the same group structure, just with different labels for their elements. For instance, the exponentiation map from the real numbers under addition
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to the positive real numbers under multiplication is an isomorphism. This means that these two groups have the same structure, even though they appear different. An injective homomorphism is called a monomorphism. Injective means, that different elements get mapped to different images. We can look at the image of the homomorphism and get that this is a subgroup of the codomain. By restricting the homomorphism to its image,
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we obtain a bijective map, meaning the domain is isomorphic to the image. In essence, a monomorphism embeds one group into a larger group. For example, all of the symmetries of a triangle are also symmetries of a hexagon, so there is a monomorphism from the dihedral group D3 to D6. A surjective homomorphism is called an epimorphism. Surjective means that every element in the codomain has at least one preimage in the domain. In this case, we can study the kernel,
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which is the set of elements in the domain that map to the identity element of the codomain. The kernel forms a subgroup and measures how much information is lost when mapping from G to H. For example, consider the map from the integers to the multiplicative group {-1, 1}, which assigns each integer to its parity: even numbers map to 1, and odd numbers map to -1. The kernel consists of all even integers. This means that, while
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individual integer values are lost in the mapping, the function retains information about whether a number is even or odd, and how even and odd integers relate to eachother. For example the fact the odd plus odd is even, is translated into -1 times -1 is plus one. That odd plus even is odd, is translated into -1 times 1 is -1. Now, let’s consider a general homomorphism between two groups. Every element in the
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kernel is mapped to the identity element. The remaining elements in the domain can be grouped into cosets, where elements in the same coset map to the same element in the codomain. The question is: can we define an operation on these cosets so that they themselves form a group and this map is again a group homomorphism? To answer this, let’s take any subgroup of a group and consider the set of its cosets. We attempt to define an operation by multiplying representatives of the
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cosets. The challenge is that different elements can represent the same coset, so the operation must yield the same result regardless of which representatives are chosen. A key observation is that if two elements belong to the same coset, their difference lies in the subgroup. Doing the same thing with the product and plugging n1 and n2 in, we get that this expression must be in the subgroup. This leads to the concept of normal subgroups.
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A subgroup is called normal, if the inverse of g times n times g is always in the subgroup for all elements g in the group and n in the subgroup. For example, every subgroup of an abelian group is normal and the kernel of every homomorphism is normal. The most important property of normal subgroups is that we can define a group structure on the cosets. The resulting group is called a quotient group. Together with the quotient group,
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we have a homomorphism called the canonical projection, which maps every element to its corresponding coset. The kernel is exactly the normal subgroup. Lets go back to the homomorphism. So we have found the operation which makes the left side into a group, now we can restrict the map to the image, which makes this an isomorphism. We get that the quotient group of G with the kernel is isomorphic to the image of f. This is also called the fundamental theorem on homomorphisms.
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Let’s apply this theorem to our previous examples. For the determinant homomorphism, the kernel consists of matrices with determinant 1, which is known as the special linear group. The image covers the entire codomain, so we conclude that the quotient of the general linear group by the special linear group is isomorphic to the multiplicative group of the field. For the exponentiation map, the kernel is the set of all integer multiples of 2 pi times i, so we get this isomorphism.
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For the last homomorphism we considered, the image is the subgroup generated by g. If no power of g equals the identity, then the kernel is trivial, and the subgroup is isomorphic to the integers. Otherwise we let k be the smallest positive integer such the g to the power of k is equal to the identity element. Then the subgroup generated by g is isomorphic to the quotient of the integers by the multiples of k.
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This also allows us to classify all cyclic groups. We get that every infinite cyclic group is isomorphic to the integers and any finite cyclic group is isomorphic to some quotient of the integers. For example the group of -1 and 1 under multiplication is isomorphic to the quotient of the integers by the even integers. More generally the group of the roots of unity of order n is isomorphic to the quotient of the integers by the multiples of n.
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Now, let’s take a closer look at the quotient group of integers. Consider the arithmetic of a clock: adding 15 hours to 3 o’clock results in 6 o’clock. We express this as 18 being congruent to 6 modulo 12. The quotient group of integers modulo 12 captures this cyclic behavior. More generally, two integers are equivalent modulo n if their difference is a multiple of n. Another important type of homomorphism is an endomorphism, where the domain and codomain
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are the same group. A bijective endomorphism is called an automorphism. The set of all automorphisms of a group forms another group under composition, known as the automorphism group. A special class of automorphisms are the inner automorphism. Every element corresponds to one inner automorphism, and we have a homorphism mapping every element to the corresponding inner automorphism. The identity element of the automorphism group is the identity map,
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the kernel is the set of all elements whose inner automorphism is the identity map. After rewriting this, we get that the kernel of the homomorphism is exactly the center of the group. So by the fundamental theorem of homomorphisms, we get that the quotient of G by the center is isomorphic to the group of all the inner automorphisms. Say we want to color the edges of a hexagon with an equal number of green and red edges.
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By applying some symmetry of a hexagon to a colouring, we get a different colouring. We can do the same thing to all of the possible colourings and obtain a permutation of all the colouring. We say that the group D6 acts on the set of all the colourings. This is an example of a group action. We define an action of a group G on a set X to be a homomorphism from the the group G into the group of all the permutations of the set X. So for every
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element in the group we have some permutation. Instead of writing the pi where the index is the group element, we just write g times x, so the group element g acts on the set element x. Equivalently, we could define a group action to be a function taking an element from the group and an element from the set and giving another element from the set, such that the identity element leaves every element unchanged, and it doesn't matter whether we first
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let h act on x and then g act on the result or we multiple g and h and let the product act on x. The second property, looks like this in practice. Here we can first rotate and then reflect, or we just perform one reflection, which is the product of the other two symmetries. Lets look at some examples of group actions. The general linear group acts on the set of n-dimensional vectors by matrix-vector multiplication. The symmetric group acts on the set of integers from 1 to n,
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where we just plug x into the permutation pi. Any group acts on itself by conjugation, where the result of g acts on x is g times x times the inverse of g. Lets get back to our colourings. We might want to know how many colouring there are. Using basic combinatorics, we get that there are in total 20 colourings. How some colourings are very similar. For example these two colouring only differ by a rotation, and these two only by a reflection. So
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we might be interested in the question of how many colourings are up to an action of D6. To make this more precise, we look at the orbit of a colouring. This is the set of all the colourings, which differ only by an action of a symmetry. So instead, we could ask how many orbits there are. Lets make these definitions more formal. So consider a group acting on some set and an element x in that set. Then the orbit of x is the set of all elements that
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can be obtained by applying elements of G to x. To understand orbits better, lets look at the map, which maps each element of the group to the corresponding element in the orbit. Then we can group together all the element which get mapped to the same element, and we get the stabilizer subgroup and its cosets. Since we now have a bijection, this means that the index of the stabilizer subgroup is equal to the number of elements in the orbit.
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We define the stabilizer subgroup to be the set of all elements which don't change x. Then the result, that the index of the stabilizer subgroup is equal to the number of elements in the orbit, is called the Orbit-stabilizer theorem. Using Lagrange theorem, we can also rewrite the equation like this. For example, when the symmetric group acts on set of integers from 1 to n, the orbit of any element is the entire set, meaning this is a transitive action. Applying
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the Orbit-Stabilizer Theorem, we find that the stabilizer of an element has size (n-1)!.
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Since the remaining (n-1) elements can be freely permuted, this confirms our intuition that the stabilizer of any element is isomorphic to the symmetric group of n-1 elements. When we look at all the possible colourings, we see that the orbits partition the set. It is now convenient to choose one representation element from each orbit. We can do the same thing for every group action. Since this the union is disjoint,
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we get that the number of elements in X is the sum of the elements in the orbits. With the orbit stabilizer theorem, we get the orbit decomposition formula. A special role play the elements of X, which are fixed by every group element. The stabilizer of these elements is the entire group and the orbit is just that element. So we can collect all the terms in the sum corresponding to fixed points, to obtain this useful formula.
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Lets look at the example, where a group acts on itself by conjugation. Here the Orbits are called conjugacy classes, the stabilizers are called centralizers and the set of all fixed points is exactly the center of the group. Applying the orbit decomposition formula, we derive the class equation of a group. Lets again turn to the question of how many orbits a group actions has. We define the set of all
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orbits. We can express the number of elements in the set by summing the reciprocal of the number of elements in the orbit for each element in the set. Using the orbit stabilizer theorem, we can express this sum using stabilizers. The sum of all the stabilizers is the same as the number of elements of this sets, which in turn is equal to the sum of the number of fixed points a given group elements has, where we are summing over all the group elements.
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Putting everything together we obtain burnside's lemma. Lets apply this, to find the number of distinct colourings. There are two fixed points of a rotation by 120 degree. For this reflection, there are four fixed points. In general, the number of fixed points for each symmetry is as follows. Now we can apply Burnsides lemma to find that up to an action of D6, there are in total three distinct colourings. The orbits look like this.
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We start with two definitions. For a prime p, a group is called a p group, if the order of that group is a power of the prime. A subgroup is called a Sylow p-subgroup if its order is the largest power of p dividing the order of the group. In this chapter, we will study p-groups and how they appear as subgroups in general groups. Many of these results rely on group actions. Suppose a p-group acts on a finite set. Using the orbit decomposition formula, we analyze the number
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of fixed points. For any non-fixed element, its stabilizer is a proper subgroup, meaning the orbit size is greater than one and divides the order of G, which is a prime power. Since the only divisors of a prime power are other prime powers, each orbit size is divisible by p. Reducing modulo p, we conclude that the number of elements in X is congruent to the number of fixed points modulo p. Applying this to conjugation, we see that the order of G is congruent to
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the order of its center modulo p. But G is a p-group, so the left side is equivalent to zero. Since we know that the center of a group is not empty, as the identity element is always in the center, we get that the order of the center is a multiple of p. Finally this means that the center of a non-trivial finite p-group is non-trivial. Another result, we could prove using this result, is that if a prime p
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divides the order of a group, then there is some element in the group of order p, meaning that the order of the subgroup generated by that element is equal to p. The Sylow theorems make statements about Sylow p-Subgroups inside of a finite group. We have that there always exists a Sylow p-subgroup, we call P. For any p-subgroup, a conjugate of that subgroup is inside of P and in particular every two Sylow p-subgroups are conjugates. And finally
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the number of Sylow p-Subgroups divides m and is equivalent to one modulo p. If the number of Sylow p-subgroups is equal to one, this Sylow p-subgroup must be normal. We can use these theorems, to classify some small groups. We already know that every group, whose order is a prime is isomorphic to some quotient of a the integers. Lets now look at groups whose order is the product of two different primes. Using the Sylow theorems, we get the following
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statements about the number of Sylow subgroups. Since p and q are primes, the only two divisors are one and the respective prime. Since p is smaller than q, p can not be equal to 1 modulo q, so we know that n q is equal to 1. For n p, the value can be either 1 or q, but if we assume that p does not divide q minus one, we guarantee, that n p is also equal to one. So both Sylow subgroups are normal and of prime order, so cyclic. We call the respective generators x and y.
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We can now form this product, which is called the commutator of x and y. On the one hand by the normality of the subgroup generated by x, we get that this product is in the subgroup generated by x, on the other hand we get that the product is in the subgroup generated by y. So it must be in the intersection. But the intersection is only the identity element, so the commutator is equal to the identity element, or equivalently x and y commute.
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We can apply this to see that the powers of the product of x and y is equal to the product of the respective powers. Using this, we can see that the order of the element the product of x and y is equal to p times q, so the group G is generated by the product of x and y. This means, that we have a cyclic group, which we already know is isomorphic to some quotient of the integers. For example this implies that the only group of order 15 up to isomorphism is the integers modulo 15.
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We were able to do all of this, since we assumed that both Sylow subgroups were normal. More generally a finite group, such that every subgroup is normal is called a nilpotent group. Since in an abelian group every subgroup is normal, an abelian group is nilpotent. To understand nilpotent groups, we define the direct product of two groups. Here we define an operation on the cartesian product of the two groups by multiplying componentwise.
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If we have a nilpotent group and have all of its Sylow subgroups, then the group is isomorphic to the direct product of its Sylow subgroups. For abelian groups, we can make an even stronger statement, as here the group is always isomorphic to some direct product of the integers modulo some prime power. This is known as the fundamental theorem of finite abelian groups.
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