Group Theory

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What are the four axioms of a group?: closed under operation associative identity element inverse elements
What is the order of a group?: It\'s cardinality, the number of elements.
What is Sn?: The set of permutations of a set with n elements. It\'s a group.
What is D2n?: The group of symmetries of a regular n-gon.
What is the order of an element?: The least positive integer such that g^n=1.
What is V?: The Klein-4 group, in which every element is it\'s own inverse.
What does it mean for a permutation to fix i?: p(i)=i
What does it mean for a permutation to move i?: p(i)=/= i
What is supp(p) the support of p?: The set of letters moved by p.
What is the order of Sn?: n!
What does it mean for 2 permutations to be disjoint?: supp(p) n supp(q) = 0 (the empty set)
When do permutations commute?: When they are disjoint.
What is the order of an r-cycle?: r
What is the order of a permutation?: The LCM of the lengths of the cycles in it\'s disjoint cycle notation.
If p is even...: It fixes the product(1=
If p is odd...: It moves the product(1=
If p can be expressed as the product of r transpositions, what is sign(p)?: (-1)^r
sign(pq)=?: sign(p)sign(q)
sign(p^-1)=?: sign(p)
What is An?: The set of all even permutations in Sn. It\'s a group.
What is the order of An?: n!/2
What 2 conditions must a subset H of a group G satisfy to be a subgroup?: closed inverses
What is if a is in G?: The cyclic subgroup of G generated by a. The smallest subgrou of G containing a.
If a group G is cyclic...: There is an a in G which generates G. G is abelian.
If H and K are subgroups of G what else is?: HnK
If a group has finite order, it\'s elements have...: finite order.
If a group has infinite order, it\'s elements...: may have finite or infinite order.
What is the order of e?: 1
If a has order n and a^m=1 then...: n|m
If a has order n then has order...: n
If has order n then a has order...: n
What is an isomorphism from (G,*) to (H,.)?: A bijection S:G->H such that S(a*b)=S(a).S(b) G=~H
If G and H are isomorphic, what is |H|?: |G|
If G is isomorphic to H then S(eG)=?: eH
If G is isomorphic to H then |S(g)|=?: |g|
What is isomorphic to Zn?: A cyclic group of order n.
What can you say about a subgroup of a cyclic group?: It\'s cyclic. Order divides n. For each divisor of n, there is 1 subgroup of this order.
What three axioms must an equivalence relation satisfy?: x~x (reflexivity) x~y => y~x (symmetry) x~y&y~z => x~z (transitivity)
An example of an equivalence relation is...: congruence modulo n.
What is a partition of a set X?: A collection of disjoint subsets of X whose union is X.
What is the name for disjoint subsets which partition X?: equivalence classes
What is the left coset of H in G determined by a?: aH={ah|h in H}
If H=: b=ah for some h in H b is in aH (a^-1)b is in H
Define a relation a~b if aH=bH. It\'s an...: equivalence relation
The left cosets of H _ G.: partition
Any 2 left cosets of H are either...: equal or disjoint.
What is |G:H|?: The index of H in G. The number of disjoint left cosets that whose union is G.
|aH|=?: |H|
Lagrange\'s Theorem: G finite: |G|=|G:H|.|H| (order of H divides order of G)
Let H=: |G:H|
If g is in G, what is the relation between their orders?: order of g divides |G|
If g is in G, g^|G|=?: 1
If G has prime order the only subgroups of G are...: {1} and G.
If G has prime order p then...: G is cyclic. Isomorphic to Zp.
Fermat\'s Little Theorem: if p is prime and p doesn\'t divide a: a^(p-1)=1 mod p
G acts on a set X if...: There is a map GxX->X, (g,x)->gx with (gh)x=g(hx) 1x=x for all x in X.
What is the orbit of x?: Gx={gx|g in G} subset of X.
The orbits of X...: partition X.
What is the stabiliser of x in G? (x is in X): G*=StabG(x) ={g in G|gx=x} subset of G.
When is a group action transitive?: When there is only one orbit.
What is left regular action?: Where G acts on itself by left multiplication. Transitive, stabiliser is {e}.
Orbit Stabiliser Theorem: G finite acting on X, x in X. |Gx|=|G|/|G*|
Cauchy\'s Theorem: Let G be finite, p a prime dividing |G|. Then G has an element of order p.
What is Fix(g)?: {x in X|gx=x} fixed point set of g
If y=gx then what are G*y and |G*y|?: G*y=(g)G*x(g^-1) |G*y|=|G*x|
Burnside\'s Lemma: Let G be finite acting on a finite set X. Then: no of distinct orbits of X under G =1/|G|.SUM(g)|fix(g)|
When a conjugate to b?: (x^-1)ax=b. x is the conjugating element.
Conjugacy is an...: equivalence relation
If A is an abelian group, how many elements are in each conjugacy class?: 1
x->g^-1xg is an...: isomorphism.
Conjugate elements have the same...: order
1 conjugated by x x conjusated by x x conjugated by x^m m in Z: 1 x x
What is the cycle type of a permutation?: If p is in Sn and p=p1...pk as a product of disjoint cycles of lengths l1,...,lk. Then [l1,...,lk] is the cycle type of p.
2 elements of Sn are conjugate if and only if they have the same...: cycle type.
What is the centraliser of a belonging to G?: CG(a)={g inG:(g^-1)=a}
CG(a) is a _ of G.: subgroup
Conjugation defines an action of G on itself. Under this action orbits and stabilisers are...: conjugacy classes and centralisers.
Let a be in G. What is the size of the conjugacy class of a?: The index of CG(a) in G.
What is the centre of a group?: The set of all x in G which commute with every element of g. Z(G)={x|(g^-1)xg=x for all g in G}
What is the centre of an abelian group A?: A
Z(G) is a _ of G.: subgroup
Let p be a prime. When is a finite group G a p-group?: Iff |G|=p^n n in N. Iff every g in G has order a power of p.
Class equation: Let G be a group partitioned into conjugacy classes. Let S be a subset of G containing exactly one element from each conjugacy class if the size of that class is more than 1. |G|=|Z(G)|+Sum(xinS)|G:CG(x)|
If G is a p-group, what can you say about Z(G)?: non-trivial.
If H is a subgroup of G, what is H^g?: {(g^-1)hg|h in H} subgroup of G.
When is a subgroup H of G a normal subgroup?: If H^g=H for all g in G. Write H triangle G.
If H is a subgroup of G, what 2 things are equivalent to \"H is a normal subgroup of G\": (g^-1)hg in H for all h in H, g in G. gH=Hg for all g in G.
Let H and K be subgroups of G. If either H or K is normal, what can you say about HK?: It\'s a subgroup of G.
Let H and K be subgroups of G. What 3 conditions must be satisfied for G to be the internal direct product of H and K, G=HxK?: H and K are normal subgroups of G. HnK={1} G=HK
If G=HxK is a direct product, what is hk?: hk=kh for all h,k. G=HK=KH.
What is the external direct pruduct of H and K (groups)?: (HxK)= {(h,k):h in H,k in K} is a group.
Let G=HxK be finite and let g=(h,k) in G. What are |g| and |G|?: |g|=lcm(|h|,|k|) |G|=|H|.|K|
Internal direct product theorem: Let G,H and K be groups. G is isomorphic to the external direct product of H and K iff G is the internal direct product of subgroups isomorphic to H and K.
When is Zn x Zm isomorphic to Zn.m?: iff gcd(n,m)=1
Define a (group) homomorphism.: A map f:G->H between groups where f(xy)=f(x)f(y) for all x,y in G.
Let f:G->H be a group homomorphism. What are f(1G) f(x^-1): f(1H) f(x)^-1
If f:G->H is a group homomorphism then then what is f(G)?: f(G)=im(f) It\'s a subgroup of H.
If f:G->H is a homomorphism, what is K={x in G|f(x)=1H}?: K=ker(f) It\'s a normal subgroup of G.
What is an epimorphism?: A surjective homomorphism.
What is a monomorphism?: An injective homomorphism.
What is a bijective homomorphism?: An isomorphism.
If f is a homomorphism, then f is a monomorphism iff...: ker(f)={1G}
If f is a homomorphism, then f(a)=f(b) iff...: aK=bK
What is G/H?: The set of left cosets of H in G.
G/H with the operation (xH)(yH)=(xy)H is called the...: quotient group or factor group.
What is the map p:G->G/H x->xH? What is it\'s kernel.: An epimorphism, the natural projection. ker(p)=H.
First isomorphism theorem: Let f:G->H be a group homomorphism. Then G/ker(f)=~ im(f). If f is an epimorphism G/ker(f)=H.
Cayley\'s Theorem: Every finite group G is isomorphic to a subgroup of Sn for some n.
Let G be a group and let H be a subgroup of Z(G). What does G/H being cyclic imply?: G is abelian. G/Z(G) cyclic => G abelian.
Let p be prime. What can you say about any group of order p^2?: It\'s abelian.
Second isomorphism theorem: Let G be a group, H a subgroup of G, N a normal subgroup of G. Then HN is a subgroup of G, HnN is a normal subgroup of G and HN/N is isomorphic to H/(HnN).
If G is the internal direct product of H and K, what can you say about G/H?: G/H is isomorphic to K.
What is a Sylow-p-subgroup of G?: If |G|=(p^k)m (p prime not dividing m) then it\'s a subgroup of G of order p^k.
What is Sylp(G)?: The set of all Sylow-p-subgroups of G.
Sylow\'s Theorems: Let G be a finite group and let p^k be the largest power of prime p dividing |G|. (1)Every p-subgroup of G is contained in a subgroup of order p^k. In particular, Sylow-p-subgroups exist. (2)Let np denote the number of Sylow-p-subgroups in G. Then |Sylp(G)|=np=1 modp (3)Any 2 Sylow-p-subgroups are conjugate
If H is a subgroup of G, what is the normaliser of H in G?: NG(H)={g in G|H^g=H}
What is a simple group?: A group with no proper non-trivial normal subgroup.
Let p and q be primes with p: They are cyclic.
Let p be a prime. What can you say about groups of order 2p?: They are either cyclic or isomorphic to D2p.

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