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Definition
The image of a function f from G to H is defined to be
im f = {f(a) | a is in G} |
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Definition
The kernel of a function f from G to H is defined to be
ker f = {a in G | f(a) = e} |
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| First Isomorphism Theorem |
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Definition
If f is a function from G to H,
(1) im f is a subgroup of H,
(2) ker f is a normal subgroup of G,
(3) im f is isomorphic to G / ker f. |
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Definition
| Let G be a finite group such that p|o(G) for some prime p. Then G contains at least one element of order p. |
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| Cauchy's theorem for finite abelian groups |
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Definition
| Suppose G is a finite abelian group and that d divides o(G). Then G has a subgroup of order d. |
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| Suppose that G is a finite abelian group such that p^a is a divisor of o(G). Then the following hold... |
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Definition
(1) G has at least one subgroup of order p^a.
(2) If p^a||o(G), then there exists only one subgroup of order p^a. |
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| If G and H are groups, the order of the group GH is... |
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Definition
| o(GH) = o(G)o(H) / o(G intersect H) |
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Definition
| Let G be a group and p be a prime such that p^a||o(G). Then a subgroup of G with order p^a is called a Sylow p-subgroup. |
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| Automorphism group, Aut G |
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Definition
Aut G = {Bijective multiplicative functions from G to G}
Aut G is a group under the composition of functions. |
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Definition
The center of a group, denoted z(G), is defined by
z(G) = {x in G | xg = gx for all g in G}
z(G) is a normal subgroup of G. |
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| Center of an abelian group |
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Definition
| A group G is abelian if and only if G = z(G). |
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| Set of inner automorphisms |
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Definition
Inn G = {a_x | x in G}
where a_x is an automorphism of G defined by a_x(g) = x^(-1) g x.
Inn G is a subgroup of Aut G. |
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| Symmetric group of degree n |
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Definition
| The symmetric group of degree n, denoted S_n, is the group of all permutations of n elements. |
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Definition
| ...commute with each other. |
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| What is the inverse of the cycle (1 2 3 4)? |
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Definition
| (1 2 3 4)^(-1) = (4 3 2 1) = (1 4 3 2) |
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Definition
| A transposition in S_n is a single 2-cycle. |
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Definition
| A permutation is called even if it is the product of an even number of transpositions. |
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Definition
| The group of all even permutations in S_n is called the alternating group A_n. Since A_n has index 2 in S_n, it follows that A_n is a normal subgroup of S_n. |
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| What is a quick way to tell if a permutation is even? |
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Definition
| A permutation is even precisely when, written as a product of disjoint cycles, there are an even number of cycles of even length. |
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Definition
Let G be a group with elements a, b. We say that a and be are conjugate if there exists a g in G such that
b = a^(-1) g a
Conjugacy is an equivalence relation on G, hence elements fall into conjugacy classes. |
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Definition
The conjugacy class of a in G is defined by
C_a = {g^(-1) a g | g in G}
It contains all elements in G that are conjugate to a. |
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Definition
| C_a = {a} if a is in the center of G. |
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Definition
Let G be a group and let a be an element of G. The centralizer of a in G is defined by
N_G(a) = {g in G | g^(-1) a g = g}
It follows that N_G(a) is a subgroup of G and that the index of N_G(a) in G is the order of the conjugacy class of a. |
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| The order of the conjugacy class of a in G is given by... |
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Definition
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Definition
o(G) = o(z(G)) + sum_a o(G)/o(N_G(a))
where o(G)/o(N_G(a)) > 1.
The sum runs over one element in each conjugacy class with cardinality 2 or greater. |
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Definition
| If G is a finite group and p is prime, a Sylow p-subgroup of G is a subgroup of order p^a, where p^a||o(G). |
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Definition
| Let G be a finite group and p be a prime such that p^a|o(G). Then G contains a subgroup of order p^a. |
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Definition
| Let G be a finite group and p be prime. Then all Sylow p-subgroups of G are conjugate in G. |
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Definition
| Let G be a finite group and p be prime. Suppose there are k_p Sylow p-subgroups of G. Then k_p|o(G). |
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Definition
Let G be a finite group and p be prime. If H is a subgroup of G of order p^a (for some natural number a), then H is contained in some Sylow p-subgroup of G.
k_p is congruent to 1 modulo p. |
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