11.5.5. Let G = (Z/24Z, +). (a) Find a subgroup H of G that is isomorphic to (Z/4Z, +). How many choices for such an H are there? (b) Find a familiar group that is isomorphic to G/H. (c) Find a subgroup K of G/H of order 3. (d) Find a familiar group that is isomorphic to (G/H)/K.

Q: 2.26. If G is a group and H is a subgroup of index 2 in G, then prove that H is a normal subgroup of…

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Q: 5. In each of parts (a) to (c) show that for the specified group G and subgroup A of CG (A) = A and…

A: (a) We are given that G=S3A=1,1 2 3,1 3 2 Now, we can see that 1 21 2 31 2-1=1 21 2 31 2=1 3 2 and 1…

Q: Find the (right) cosets for the following subgroups of the stated groups: (i) The subgroup H =…

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Q: 10.3.3. Let G Z/4Z x Z/6Z, and let H be the subgroup of G generated by (2, 2). (a) What are the…

A: Given that G=ℤ/4ℤ×ℤ/6ℤ.i.e. G=ℤ4×ℤ6H is a subgroup of G generated by 2,2i.e. H=<2,2> oG=4×6=24

Q: 2. First Isomorphism Theorem. Let G be a group and H-((.s)|ge G). (a) (b) (1) G is abelian. (i) H is…

A: Given G is a group then, G×G is a group of the type such that G×G={(a,b); where a, b belong to G}…

Q: 2. Let G be the group of 2 x 2 matrices under addition and H = {(ab): a +d=0}. Prove that H is a…

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Q: 14*. Find an explicit epimorphism from S4 onto a group of order 4. (In your work, identify the image…

A: A mapping f from G=S4 to G’ group of order 4 is called homomorphism if :

Q: 22. Let 0 G → H be a group homomorphism, and let K = ker = {a € G|0(a) = EH}. Show that gK = Kg for…

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Q: 4.6. (a) Let G be a cyclic group of order 6. How many of its elements generate G? Answer the same…

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Q: (b) Let G be a group and let x E G. Suppose that (x) is a cyclic group of order N. Let m and n be…

A: We make use of the gcd to answer this question. DEtails are given in step 2.

Q: 6.9. Let G = Cn be a cyclic group of order n, and let X = {x₁,...,n} be a set containing n elements.…

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Q: Which of the following sets forms a subgroup in the additive group Z? (a) {1} (b) N (c) {1, -1}. (d)…

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Q: (a) Make addition and multiplication tables for Z/3Z.

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

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: uppose that at 12:01 a.m. on January 1 (just after midnight), there is one cell. And suppose it…

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Q: 7. (a) What is the order of (1, (123)) in Z4 × S3 ? (b) Write down all the distinct cosets of the…

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Q: 12.3.1. Let G be a group of order 585 = 3² × 5 × 13. (a) Prove that G is not simple. The group G…

A: Since you have posted a question with multiple sub parts, we will  provide the solution only to the…

Q: 3. Use the three Sylow Theorems to prove that no group of order 45 is simple.

A: Simple group: A group G is said to be simple group if it has no proper normal subgroup Note : A…

Q: 4

A: We use theorem of order of group and order of elements of group .

Q: the group (a) find all possible orders of elements in G; (b) for each possible order of an element…

A: (ℤ/364ℤ,+) is isomorphic to (ℤ364,+). All possible orders of (ℤ364,+) are all possible divisors of…

Q: Hello, can you help me out with this problem? Please write a solution on a piece of paper and upload…

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Q: 4. Consider the subgroup H = {1, 10, 26} of U37. (a) Find and compare the orders of 6 E U37 and 6H E…

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Q: 5. Let G be a group. Let H₁ CH₂ C H 3 C · of subgroups of G, and let H∞ = Un=1 Hn. be an infinitely…

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Q: Let G be a group of order 40 whose subgroups include H and K, of orders 4 and 5, respectively. to…

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Q: (4) Let G be a group. Prove that Z(G) is a normal subgroup of G.

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Q: Consider the group A₁. Let H = {e, (12)(34), (13)(24), (14)(23)}. (a) Show that H is a subgroup of…

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