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42 Cards in this Set
- Front
- Back
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Theorem 4.1:
Let G be a group and a{G. If a has infinite order, then a^i=a^j IFF... if a has finite order of n, then a^i=a^j IFF... |
... i=j
... n divides i-j |
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Let a be an element of order n and let a^k=e. Then we know...
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... n divides k
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Let a be an element of order n and let k be a positive integer. Then <a^k>=...
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...<a^gcd(n,k)>. And the order of a^k is n/gcd(n,k)
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In a finite cyclic group, the order of an element...
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...divides the order of the group
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The order of a is n. Then <a^i>=<a^j> IFF...
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...gcd(n,i)=gcd(n,j)
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Let the order of a be n. Then <a>=<a^j> IFF...
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...gcd(n,j)=1
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What is the fundamental Theorem of Cyclic groups?
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(a) Every subgroup of a cyclic group is cyclic
(b) If the order of <a> is n, then the order of any subgroup of <a> is a divisor of n (c) For each positive divisor k of n, the group <a> has exactly one subgroup of order k, namely, <a^n/k> |
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If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is...
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... U(d) (or the number of integers less than d that are relatively prime to d)
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A permutation of a set A is a function...
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...from A to A that is both one to one and onto
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A permutation group of a set A is...
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...a set of permutations of A that forms a group under function composition
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Every permutation of a set can be written...
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...as a cycle or as a product of disjoint cycles
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If the pair of cycles (a1,...,aN) and (b1,...,bN) have no entries in common, then...
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... ab=ba
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The order of a permutation of a finite set written in disjoint cycle form is...
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...the least common multiple of the lengths of the cycles
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An isomorphism from a group G to a group G' is...
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... a one to one mapping (or function) from G to G' that preserves the group operation
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What are the 4 steps in proving a group G is isomorphic to a group G'?
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(1) "Mapping"; define a function from G to G'
(2) "1-1"; Prove that the function is one to one; that is, assume X(a)=X(b) and prove a=b (3) "Onto"; Prove that X is onto; that is, for any element in the range, find an element in the domain that maps to that element (4) Preserves operation. Show that X(ab)=X(a)X(b) for all a and b in G |
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Cayley's Theorem: Every group is isomorphic...
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...to a group of permutations
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A binary operation is...
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...a function that assigns to each ordered pair of elements of a set G an element of G
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Let G be a set together with a binary operation. We say G is a group if...
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...(1) The operation is associative. (ab)c=a(bc)
(2) There is an element e that is the identity (3) For each element in G, there is an element that is an inverse |
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The identity in a group is...
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...unique
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In a group G, do the right and left cancellation laws hold?
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Yes, always
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For each element a in a group G, there is a unique element b such that...
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...b is the inverse of a
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Let G be a group. What is the order of G?
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The order is the number of elements in G
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The order of an element g in a group is...
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...the smallest positive integer n such that g^n=e
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If a subset H of a group G is itself a group under the operation of G, then...
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...we say H is a subgroup of G
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One-Step subgroup test: Let H be a nonempty subset of group G. If a,b are in G, then...
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...H is a subgroup is ab^(-1) is always in H
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Two-step subgroup test: H is a nonempty subset of G. a and b are in H. If...
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...ab is in H and a^(-1) is in H, then H is a subgroup
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Finite subgroup test: Let H be a nonempty finite subset of G. H is a subgroup if...
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...it is closed under the operation of G
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The cyclic set of any element in a group is..
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...a subgroup
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The center, Z(G), of a group is...
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...the subset of all elements in G that commute with ALL elements of G. The center is a subgroup
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The centralizer of a in G, denoted C(a), is...
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...the set of all elements in G that commute with a, and is a subgroup
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What are the first 4 properties of Isomorphisms acting on elements. (Let R be the isomorphism) of a group G onto G'
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(1) R carries the identity of G onto the identity of G'
(2) For every integer n and every group element a, R(a^n)=R(a)^n (3) For any elements a and b in G, a and b commute if and only if R(a) and R(b) commute (4) G=<a> IFF G'=<R(a)> |
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What are the last 3 properties of Isomorphisms acting on elements. (Let R be the isomorphism) of a group G onto G'
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(5) |a|=|R(a)|
(6) For a fixed int. k and fixed element b, the equ. x^k=b has the same # of sol's in G as does x^k=R(b) in G' (7) If G is finite, then G and G' have exactly the same number of elements of every order |
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What are the 4 properties of isomorphisms acting on groups? (Let R be an isomorphism from G to G')
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(1) R^(-1) is an isomorphism from G' onto G
(2) G is Abelian IFF G' is abelian (3) G is cyclic IFF G' is cyclic (4) If K is a subgroup of G, then R(K)={R(k): k<K} is a subgroup of G' |
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An isomorphism from G onto itself is...
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...an Automorphism
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Let a be in a group G. The function R(sub)a defined by R(sub)a[x]= axa^(-1) for all x in G is...
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...called the inner automorphism of G induced by a
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The set of automorphisms of a group and the set of inner automorphisms of a group are...
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...both groups under the operation of function composition
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For every positive integer n, Aut (Zn) is isomorphic to...
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... U(n)
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The intersection of two subgroups is...
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...a subgroup itself
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Lagrange's Theorem says...
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...If G is a finite group and H is a subgroup of G, then |H| divides |G|. Moreover, the number of distinct left (right) cosets of H in G is |G|/|H|
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In a finite group, the order of each element...
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...divides the order of the group
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A group of prime order...
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...is cyclic
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Let G be a finite group and a be in G. Then a^(|G|)=...
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...e
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