The number of normal subgroups of a non trivial simple  group is  Select one: a. 2 b. 3 c. 1 d. 0

Q: The following is a Cayley table for a group G, 2 * 3 * 4 = 3 1 2. 4 主 3. 4 2 1 21 4 345

A: For group, 2*3*4=(2*3)*4.

Q: Let G= How many elements does G have?  To what familiar group is G isomorphic?

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Q: Which of the following is group under multiplication? O a. Q-{0} O b. None of them О с. (-{1} O d.…

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Q: H and K are subgroup of a group G, then which of the following is a group? a) HUK b) HK c) HnK d)KH.

A: Subgroup: Let (G,*) be a group and H be a non-empty subset of G; then, H is called subgroup if H…

Q: Q/ How many non-trivial subgroups in s, ? a) 2 b) 3 c) 4

A: S3 is the set of permutations on the set {1, 2, 3}S3={e, (1 2), (1 3), (2 3), (1 2 3), (1 3 2) }The…

Q: Determine whether the following is a semigroup, monoid, or a group. 1. G=Z, a*b = a + ab 2. G=2Z,…

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Q: C. Find the number of elements in the indicated cyclic group. 1) The cyclic subgroup of Z30…

A: Given: The cyclic subgroup of 230 generated by 25. To find the number of elements that is indicated…

Q: set G be a finite group, P E Syl,(G), and N = (a) Show that Pe Syl, (N). Jumber and Conjugacy of…

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Q: Find the cyclic subgroup of S6 generated by the element (123)(456)

A: If G is a group and g is an element of G of order n, then the cyclic subgroup of G generated by g is…

Q: Give the subgroup diagram for each of the groups: (a) Z24 (b) Z36-

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Q: (a) In S4, find the subgroup H generated by (123) and (23) (b) For o = (234), find the subgroup oHo

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Q: In the following Cayley table for a group G, C(3}= 2 3 4 3 4 5 4 1 3 4 1 2 3 2 3 4 4 2 4 O {1,2,4} G…

A: If G be a group and a belongs to G then c(a) = { x : ax = xa , where x belongs in G}.

Q: Let G be a group of order n. For what value of n is the group G NOT cyclic?

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Q: In the following Cayley table for a group G, C(2)= 2 2 TIT

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Q: For a cyclic group of order 420, the number of generators are, (a) 66 (b) 76 (c) 86 (d) 96

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Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

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Q: Consider the group (Z,*) defined as a*b=a+b , then identity (Neutral) element is     a 1   b -1…

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Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.

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Q: Q/ How many non-trivial subgroups in s3 ? a) 2 b) 3 c) 4

A: We know that S3 = (1) , (1,2) , (1,3) , (2,3) , (1,2,3) , (1,2,3) Thus the subgroups of S3 are given…

Q: Let G = A4, the alternating group of degree 4. (a) How many elements of order 3 does G have? (b)…

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Q: The following is a Cayley table for a group G. 2*5 4 = 1 1 3 1 4 2 4 4. 4. 5. 4. ENG pe here to…

A: 1

Q: 50. How many proper subgroups are there in a cyclic group of order 12? A 4 в з с 2

A: see 2nd step

Q: How many subgroups of the group G = (Z30, +30)? O a. 4 O b. 7 O c. 5 O d. 6 O e. 8

A: We know that τ(n) is number of positive divisors of n Also τpaqbrc = (a+1)(b+1)(c+1) , where p , q ,…

Q: order 8 of a group G, and =

A: Given that order of a is 8 .Then  a8=e Rearrange  a little bit , we can have a42=e Hence order of…

Q: C. Find the number of elements in the indicated cyclic group. 1) The cyclic subgroup of Z30…

A: 1 25 ∈ Z30    ∴ 025=01ged25,30               =305=6    ∴25 has 6 elements  2 30 ∈ Z42…

Q: Is the set of numbers described below a group under the given operation? Whole numbers; addition ...…

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Q: Example: H.W In the group (Z6,+6) find the order of each element in

A: Explanation of the solution is given below....

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…

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Q: One of the following is a p-sylow subgroup of the group Z, × Zg H={(0,0),(0,4)} O…

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Q: 15.Consider the group (A, *) where *is defined by the table below A B C D ABA D C BA B C D C D C A B…

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Q: Is the set of numbers described below a group under the given operation? Prime numbers;…

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Q: Let p and q be distinct prime numbers and set n = pq. Find the number of generators of the cyclic…

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Q: This is Abstract algebra question: How many subgroups of order 4 does D_4 have?

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Q: The following is a Cayley table for a group G. 2* 5*4 = 1 2 3 5 2 3 4 3 4 2 3 5 1 4 2 3 4 1 2 4 1.…

A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…

Q: Let G=.  How many elements does G have?  To what familiar group is G isomorphic?

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Q: The group ((123)) is normal in the symmetry group S3 and alternating group A4.

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Q: Is the set of numbers described below a group under the given operation? Even integers; addition O…

A: A set is called group on binary operation if a) It is closed b) It is associative c) It has identity…

Q: Find the solution set for each of the following with the representation of the group on the number…

A: To find - Find the solution set for each of the following with the representation of the group on…

Q: For A, the alternating subgroup of S, show that it is a normal subgroup, write out the cosests, then…

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Q: Consider the "clock arithmetic" group (Z12,O), together with subgroup H= {0,4, 8} Write all cosets…

A: Consider the ‘clock arithmetic’ group Z12, ⊕, together with subgroup H=0,4,8. The objective is to…

Q: Suppose that N and M are two normal subgroups of a group and that IOM = {e}. Show that for any n E…

A: Given: N and M are two normal subgroups of G and N ∩ M = {e} To prove: nm = mn for any n∈ N and m∈ M

Q: Consider the alternating group A4. (a) How many elements of order 2 are there in A4? (b) Prove that…

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Q: (b) Consider the incomplete character table for a group given below:

A: The table for the incomplete character is,   1 a b c d x1 1 1 1 1 1 x2 1 1 -1 -1 1 x3 1 1…

Q: need help with cyclic groups plz, thanks

A: Given groups are,

Q: Is the set of numbers described below a group under the given operation? Integers; addition Yes O No

A: Given: A set of integers under the addition operation. We have determine whether the set of integers…

Q: Let G be a group and let a E Ga G with a = 8. the order of a² is not equal to the order of the…

A: The given statement is

Q: (b) Complete the following character table of a group of order 12: 1 3 4 X1 X2 X3 4,

A: The character table of a group of order 12:

Q: Consider the group D4 and the subgroup {I, F}. List all the left cosets of H (with all the elements…

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