3. Let G be a finite group of order n and identity element 1. a. Show that g" = 1 for any g€ G. b. An element g e G is said to be square if there exists x € G such that g = x2. Prove that G has odd order if and only if every element of g is a square. |

Q: SUCH THAT LET H BE A PROPER SUBGROUP OF G V x,y € G-H, xy EH. PROVE THAT HAG.

A:

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

Q: 9.12 Let G be a group and for elements a, b e G let a R b mean that there exists an element x = G…

A:

Q: 3. Let G be a group. Recall that if x E G, we defined the conjugacy class of x in G, denoted by…

A:

Q: Let G:- [0, 1) be the set of real numbers x with 0<x<1. Define an operation + on G by ** y={x+y if…

A: Solution:-  Given   G=0,1   be the set of real numbers xwith  0≤x&lt;1.  The operator ∗on G is given…

Q: Let r be the second largest digit in your student ID. Let G and H be groups, and let p : G → H be an…

A: Given that G and H be groups and φ : G→H be an isomorphism. Also given g∈G have order r. To show: φg…

Q: Suppose o is a homomorphism of groups fom G onto H. If H has an element of order 100, then show that…

A:

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: Let G be a group and let a be a fixed element of G. Show that H = {x E G |xa = ax} is a subgroup of…

A:

Q: Let a and b be elements of a group G. We define the conjugacy relation a-b by the following…

A: According to the given information,

Q: 1. Let A be a non-empty set and (G, .) be a group. Let F be the set of all functions from A to G.…

A: Let A be a non empty set. For f,g∈F with the operation f*gx=fx.gx To prove that F,* is a group.…

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

A:

Q: Suppose that X and Y are subgroups of G if |X|-32 and IYI=48, then what is the best possible of Xn…

A:

Q: 1. Consider the following groups G and H, K < G. If G is isomorphic to H × K, give an isomorphism v…

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: 5*. Suppose that 0: G -+ H is a homomorphism. Prove that (a) (a")= (x)" for all positive integers n.…

A: To prove the properties of group homomorphisms

Q: 2. Let G and H be groups, and let a: G H be a homomorphism. Prove the following statements. (a) a(G)…

A: Let us prove above results

Q: 2. Let G and H be any two groups. Define the set G x {e} = {(g, eμ) | gЄG} which is a subset of G ×…

A: Step 1: Step 2: Step 3: Step 4:

Q: Let G and H be finite groups. Prove the following statements. (a) If a: G H is a homomorphism, then…

A: Given that  and  are finite groups.We need to prove the following.(a) If  is a homomorphism, then…

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: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

A:

Q: Exercise 7.15. Let G be a finite group, g € G, and N≤ G. Prove that the order of gN in G/N must…

A:

Q: 3. Let G be a finite group of order |G| = n. Prove that G is isomorphic to a subgroup of Sym(G)…

A: The objective is to prove that G is isomorphic to a subgroup of symmetric group.

Q: 4. Let G be a finite group of order 4 containing no element of order 4. Explain why every…

A: See below

Q: A group G acts faithfully on a set S if g * s = s for all s € S implies g = e. Show that G acts…

A: Proof 1: If G acts faithfully on S then mg≠mg', where g≠g' ∈G. Let G acts faithfully on S. Assume to…

Q: Let G be a group, let f, g = G, and suppose that f~G 9. Show that the order of f is equal to the…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

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: 4. Let G and G be groups and let y be an isomorphism from G to G. Prove that if KCG, then y(K) CG.

A:

Q: Let G=U(20) and H={1,9} be a subgroup of G. The number of distinct left cosets of H in G is: * O 4 O…

A: We have to solve given problem:

Q: 5. True or False. Explain your answers. (a) Suppose f: G → H is a homomorphism and g € G. Then, the…

A: We have to answer that statements are true or false.

Q: et G = {fab | a, b € R, a 0}, where f(x) = ax + b for all real x. Consider the group G under the…

A: Given that G is a group defined as            G = { fa,b | a, b ∈ R , a≠ 0} .   where   fa,b(x)=ax+b…

Q: 1. Let G be the set of integers under addition and H = {-1,1} be a multi- plicative group. Show that…

A:

Q: Let G and H be two groups. Let a € G have order n and 6 € H have order m. Show that (a, b) € G x H…

A: G and H are two groups.a∈G have order n and b∈H have order m.

Q: 52. Let G be a group. Complete the following statement: |Inn(G)| = 1 if and only if

A: Let G be group. Complete the following statement: |InnG|=1 if and only if _________