Find all of the abelian groups of order less than or equal to 40 up to isomorphism.
Please show work, thank you!

Find all of the abelian groups of order less than or equal to 40 up to...
(a) State the Fundamental Theorem of Finitely Generated Abelian Groups. (b) List all abelian groups of order 2450 up to isomorphism. (c) Show every abelian group of order 2450 has an element of order 70.
(7)(20 points) (a) State the Fundamental Theorem of Finitely Generated Abelian Groups. (b) List all abelian groups of order 2450 up to isomorphism. (c) Show every abelian group of order 2450 has an element of order 70.
(7)(20 points) (a) State the Fundamental Theorem of Finitely Generated Abelian Groups. (b) List all abelian groups of order 2450 up to isomorphism. (c) Show every abelian group of order 2450 has an element of order 70.
Use the Fundamental Theorem of Finitely Generated Abelian Groups to answer the following: a. Find all abelian groups, up to isomorphism, of order p3 where p is a prime b. Use part (a) with a suitable p to list all possible abelian groups that are isomorphic to (Z2x From this list, identify the abelian group that is isomorphic to (Z2xZ8)/(1, 4))
Count the number of abelian groups (up to isomorphism) of order at most 10.
Find all non-isomorphic abelian groups of order 48
Find Aut(ℤ15). Use the Fundamental Theorem of Abelian Groups to express this group as an external direct product of cyclic groups of prime power order. Please provide as much work and explanation as is relevant.
please show your steps and your work clearly thank you!
2. Find all the subgroups of D4. Which subgroups are normal? What are all the factor groups of D4 up to isomorphism? Find all the subgroups of the quaternion group, Q8. Which subgroups are 3.
Tell whether the pH at equivalence point is less than 7, greater than 7 or equal to 7 for each of the following solutions. PLEASE SHOW WORK IF POSSIBLE OR EXPLAINATION; Thank you so much! 12. 0.1M acetic acid with 90.10M NaOH a). Greater than 7 b.) Equal to 7 c.) Less than 7 d.) Equal to 14 e.) Don't Know 13. 0.050M formic acid (HCOOH) with 0.0930M NaOH a.) Greater than 7 b.) Less than 7 c.) Equal to...
1. (a) If you recall the argument we made when finding that all simple finite abelian groups are cyclic of prime order assumed we were talking about a finite group. Show that the we reach the same conclusion without the finiteness assumption That is, show that if G is a simple abelian group then G is of prime order. (b) Show that Z has no composition series. 1101