Let 3 denote the centre of a group G. then show that a 3 iff N(a) = G. also show that if G is finite, then a € 3 iff o(N(a)) = o(G)

Q: An is the alternating group on n letters. What is |An|?

A:

Q: Build the operation table ∗ on G = {e, a, b} knowing that (G, ∗) is a group.

A:

Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.

A:

Q: Consider the factor group G = R* /Q*. Determine which of the following elements in G are equal. 1Qx,…

A:

Q: Give an example of a p-group of order 9.

A: Given, Give an example of a p-group of order 9.

Q: For each integer n > 1, give examples of two nonisomorphic groupsof order n2.

A: To Do: For each integer n > 1, give examples of two non-isomorphic groups of order n2.

Q: 3. Suppose that a group G contains elements of orders 1, 2, 3, 4, ..., 8, 9. What is the minimum…

A:

Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

Q: Let (G, *) and (H,) be groups. If group (G, *) is cyclic and & H, then (H,) is also cyclic. Select…

A:

Q: Let Zeven represent the set of even integers, {... -6, -4, -2, 0, 2, 4, 6, ...}. Determine if the…

A: (i) Closed: Yes, the set of even integers is closed under addition, meaning that when two even…

Q: (B) Let G be a set of all real numbers excep: . Define * on G by a * b = |a|b, where |a| is the…

A:

Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

A:

Q: Let the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: Let G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar groupis G isomorphic?

A: Here given that  G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar group is G isomorphic…

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: Let G, H, and K be groups. Show that (G x H) × K = G × (H × K). Hint: An element of (G × H) × K will…

A:

Q: Show, by example, that in a factor group G/H it can happen thataH = bH but |a| ≠ |b|.

A:

Q: Consider the groups  and .

A: We have to consider the groups

Q: Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

A:

Q: Determine whether the set G is a group under the operation * G= integers Z ; a*b= a-b

A: According to the given information it is required to check whether the given set G is a group or…

Q: Let G = Z4 x Z6. Compute the factor groups G/((2, 3)) and G/((3, 3)). (In each case, write the…

A:

Q: The order of identity element equal itself in any group

A: .

Q: which of the following group are cycle ?(a) U(35) (b)U5(40)

A: It is required to check which of the given group are cyclic. (a) U(35) (b)U5(40)

Q: 2. Let G = {a, b, c, d, e} with the operation * defined by the table below. Determine if the system…

A: Let G be a set and * be a binary operation then (G,*) is a group if  (G,*) has closure property.…

Q: (6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove…

A:

Q: Q2 / If a, b and c are elements of a group (G,*) such that c*a=c*b then a=b.

A:

Q: Let G = <a, b |a2 = e, b2 = e, aba =bab>. To what familiar groupis G isomorphic?

A: It is given that, G = a, b : a2=e, b2=e, aba = bab be a group. We have to find the familiar group…

Q: Let Z denote the centre of a group G. Then show that z € Z iff N(a) = G. Also show that if G is…

A: Let Z denote the centre of a group G,  we have to show that a∈Z iff N(a)=G. ,also show that if G is…

Q: Let's consider a group G with the operation * defined as follows: For any two elements a, b in G, a…

A:

Q: Determine whether the set G is a group under the operation * G={n integer|n is odd}; a*b=a+b

A:

Q: If G is a group under the binary operation (*), Then (a*b)-1=(b-1)*(a-1). Proof this statement with…

A:

Q: 3. Verify Test the properties of a group : Let G denote a set of all ordered pairs of real-valued…

A: A algebraic set is said to be the group if it satisfies the following three properties. 1)…

Q: Determine if the following sets G with the operation indicated form a group. If not, point out which…

A: A)Sol: We have G= set of all integers  and the binary operation on G is defined by a*b=a-b , where a…