Abstract Algebra Theorems and Proofs Flashcards

Questions and Answers

What does Cayley's Theorem state?

Every group is isomorphic to a group of permutations.

What does the Division Algorithm guarantee for integers m and n?

There exists unique q and r such that n=qm+r and 0 ≤ r < m.

What is stated in Corollary 8.4 regarding homomorphisms?

If h:G->G' is a homomorphism and g in G, then h(g^n)=[h(g)]^n for all n in Z.

What are the two properties of a homomorphism as described in Lemma 8.5?

h(H) is a subgroup of G' and if h is 1-1, then H is isomorphic to h(H). Signup and view all the answers

What does Theorem 9.1 indicate about elements in Sn?

There exists disjoint cycles such that σ=(μ1,...,μk) and Sn is generated by its set of cycles. Signup and view all the answers

What is the significance of Corollary 9.2?

Every permutation of Sn (n ≥ 2) can be expressed as a product of transpositions. Signup and view all the answers

What does Theorem 10.1 state about cosets and groups?

The set of left cosets of H forms a partition of G, and the set of right cosets of H forms a partition of G. Signup and view all the answers

What does Corollary 10.3 imply about groups of prime order?

If |G| is prime, then G is cyclic. Signup and view all the answers

What does Theorem 11.1 state regarding the orders of elements in groups?

|(g,h)| = lcm (|g|,|h|) where g is in G and h is in H. Signup and view all the answers

What condition must be met for GxH to be cyclic according to Theorem 11.3?

|G| and |H| must be relatively prime. Signup and view all the answers

Study Notes

Cayley's Theorem

Every group G can be represented as a group of permutations of itself.
Defined a map τ_g: G -> G by τ_g(x) = gx; it's a bijection.
If τ_g(x) = τ_g(y), then by cancellation, x = y (injectivity).
The map is onto since τ_g(hg^-1) = h for any h in G.
A homomorphism φ: G -> S_G is established where φ(g) = τ_g.
Shows that φ(gg^hat) = φ(g)φ(g^hat) indicating it's well-defined.
If φ(g) = φ(g^hat), by injectivity, g = g^hat, confirming isomorphism.

Division Algorithm

For any integers m > 0 and n, there exist unique integers q and r such that n = qm + r and 0 ≤ r < m.

Corollary 8.4

If h: G -> G' is a homomorphism and g in G, then h(g^n) = [h(g)]^n holds for all integers n.

Lemma 8.5

For a homomorphism h: G -> G' and subgroup H of G:
- h(H) is also a subgroup of G'.
- If h is injective, then H is isomorphic to h(H).

Theorem 9.1

Every permutation σ in symmetric group S_n can be expressed as a product of disjoint cycles.

Corollary 9.2

Any permutation in S_n, where n ≥ 2, can be expressed as a product of transpositions.

Theorem 9.3

If σ is expressible in two different ways using transpositions, both expressions share the same cycle pairing.

Theorem 10.1

For a group G and a subgroup H:
- Left cosets of H within G partition G.
- Right cosets of H within G also partition G.

Corollary 10.3

If a group's order |G| is prime, then G is cyclic.

Corollary 10.4

In a finite group G, the order of any element g in G divides the order of G.

Theorem 10.5

For subgroups K and H of G:
- If the indices [G:H] and [H:K] are finite, then [G:K] = [G:H][H:K].

Theorem 11.1

For groups G and H with elements g in G and h in H of finite order, the order of the pair (g, h) is the least common multiple of their individual orders, |(g,h)| = lcm(|g|, |h|).

Corollary 11.2

For groups G1,...,Gn where each gi has finite order, the collective order is given by |(g1,...,gn)| = lcm(|g1|,...,|gn|).

Theorem 11.3

The direct product G x H of finite cyclic groups is cyclic if and only if the orders |G| and |H| are relatively prime.

Corollary 11.4

The direct product of finite cyclic groups ∏_(i=1)^n G_i is cyclic if and only if the orders |G_i| and |G_j| are relatively prime for all i ≠ j.

Description

Explore key concepts in abstract algebra with these flashcards focusing on theorems, corollaries, lemmas, and their proofs. This resource is designed to help you understand the fundamental structures of groups, including Cayley’s Theorem and its implications. Perfect for students preparing for exams or wanting to solidify their knowledge in abstract algebra.