6. Let H be a subset of a group G that satisfies the following properties a. H + 0 b. if x e H andy e H, xy e H. C. if x e H then x- e H. Prove e e H.

Q: 11. Let H be a subgroup of a group G and suppose that 91,92 € G. Prove that the following conditions…

A:

Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

Q: 22. Let G be a group and let H and K be subgroups of G. For a € G, we define the double coset HaK =…

A: We show the equality of the sets by showing one is contained in the other and vice-versa.

Q: The following is a Cayley table for a group G. 2.5.4 = 2| 1 2 3 4 1 3 4 1 3 4 1 2 3 3 4 1 4 1 2 1 3…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 2. Let G be a group. Recall we showed in class that if |G| is finite, then o(g) is finite for all g…

A: be a group.If  is finite then  is also finite where We have to show the converse of the statement…

Q: 7. Let G be a group and for elements a, b e G define aRb if and only if there exists an element x E…

A:

Q: P2 find the change-of-coordinates matrices from basis B to basis C. {1−2t+t², 4+7t + 5t², 5-8t+8t²),…

A: The solution is given by

Q: For any group element a and any integer k, show that C(a) C C(a*). Use this fact to complete the…

A: To prove: Ca⊆Cak for any group element a and any integer k. Also, To Complete the statement "In a…

Q: 3. Let G be a group of order 8 that is not cyclic. Show that at = e for every a e G.

A: Concept:

Q: Show that if G is a finite group of even order, then there is an a EG such that a is not the…

A:

Q: 30. Prove the right and left cancellation laws for a group G; that is, show that in the group G, ba…

A:

Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

Q: 48. Let G be a group and g = G. Show that Z(G) = {x ≤ G : gx = xg for all g = G} is a subgroup of G.…

A: A non empty subset S of a group (G,∗) is said to be a subgroup of G if S itself a group. That is i)…

Q: Let G be a group and let a e G. The set C(a) = {x € G | xa = that commute with a is called the…

A:

Q: 1. Let G be a p-group acting on a finite set X. Let XG be the set of fixed points of this action: XG…

A: The solution lies using the Orbit Stabilizer Theorem

Q: Let X be a set with exactly n elements, where n ≥ 2, and consider the group P(X) with the symmetric…

A: The question you provided is related to concepts from abstract algebra and set theory. Here are the…

Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

A:

Q: 4. Let G be a group of order 49. Prove that G is cyclic or g7 = e for all g € G.

A: Given a group of order 49, then to prove that either G is cyclic or g7=e for all g∈G.

Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: . Let G be a group, let g e G, and let H - G. Suppose that the element Hg E G/H has order n. Show…

A: We have to prove that given statement:

Q: et G be a group and suppose that x E G has order n. Let d be a divisor of n. Show that G as an…

A: