If G and G' are both groups then G' ∩ G is a group. True or False then

Q: 5. Let G be the multiplicative group {[2], [4], [6], [8]} = Z10-

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Q: Prove that a group of order 28 must have an element of order 2.

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Q: group of order 126 cannot be simple

A: a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group…

Q: Prove or Disprove: If (G, *) be an abelian group, then (G, *) a cyclic group?

A: If the given statement is true then we will  proof the statement otherwise disprove we taking the…

Q: For any two elements a and b of a group G is abelian iff (ab)² = a²b². , show that G

A: For any two elements a and b of a group GClaim : G is abelian iff (ab)2 =a2b2 .

Q: true O false O * -2 2) Let |G| = n and H a subgroup of G then la, + H| = la, + H| = laz * H| = ... =…

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Q: Qa\ Answer for only one. 1) Let H and K be normal subgroups of a group G, then Hn K be a normal…

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Q: If f is an isomorphism of a group G into a group H, prove that (f(a))n=e' if and only if an=e. Give…

A: Suppose f is an isomorphism from G to H, and let a be an element of G. Then: (f(a))^n = e' if and…

Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy<minox, oy. Let us take the group…

Q: If G is a group then Z(G)◄G. *True False

A: Use the relevant definitions.

Q: that a and xax^-1 have same order for all x belongs G.

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Q: Prove that  a group of order 1000 cannot be simple

A: Sylow's theorem: Let n be a positive integer that is not prime, and let p be a prime divisor of n.…

Q: Prove that the external direct product of any finite number of groupsis a group.

A: To Prove: The external direct product of any finite number of groups is a group.

Q: (4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹.…

A: Let  be  a  group  and  assume  that Suppose  that Since,  and  is  a  group, it follows  that…

Q: Determine the group Z42/ and find a group that it is isomorphic to it. Generalize your observation.

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Q: prove that Every group oforder 4

A: Give statement is Every group of order 4 is cyclic.

Q: Q1)) prove or disprove ( 1. Let (G, *) be a group, if x*y = y*x then (x*y)" = x" * y". 2. Each…

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Q: Let G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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Q: *Prove: Every subgroup of a cyclic group is cyclic.

A: Let G be a cyclic group generated by an element a. That is, G = ⟨a⟩. We want to prove that every…

Q: 2. Let G be a group. Pro-

A: Let G be a group .

Q: (4) Let G be a group, a, b E G such that |a| = 20, |b| = 12. Suppose (a) (b) ‡ {e}. Prove that b²a¹0…

A: <a>∩<b>≠e<a>∩<b> order either 2 or 4if (<a>∩<b>) has order 2then…

Q: Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e,…

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Q: Let (G, *) and (H, *) be groups. The direct product of G and H is the set G × H equipped with…

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Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

Q: 1. Show that every group of prime order is simple.

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Q: If G is a group, prove that Aut(G) and Inn(G) are groups

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Q: Q3) Prove or disprove 4. Any non-trivial group has at least 2 normal subgroups.

A: Here we have to prove that any non trivial group has atleast 2 normal subgroups.

Q: t the groups: G, G', G'' and the functions f:G-->G', g:G'-->G'' Prove the following statement: If…

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Q: set P(S)

A: According to the question, The power set P(S) of a set S is the set of all subsets so, of S,…

Q: B Suppose (G, *) is group of order > 1. Which statement best describes problem? Cannot have zero,…

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Q: 2. Which of the following are groups? For those which are not groups, explain why not (it is enough…

A: 2)(i)  under   is  not  a  group  since which has  no  inverse  under    (ii) Let      Then,   is…

Q: 1) Let G be a group and H be a subgroup of G then : a = b mod H + b-1 * H = a-1 * H . O true O false…

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Q: Let G be a group and let a, be G such that la = n and 6| = m. Suppose (a) n (b) = (ea). Prove that…

A: According to the given information, let G be a group.

Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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