You can decide whether a smaller set inside a group is itself a group, and use Lagrange's theorem on the sizes. You can also list the cosets it produces.
Apply the one-step subgroup test.
Use Lagrange's theorem: a subgroup's order divides the group's order.
List the left cosets of a subgroup of order 3 inside a group of order 12.
Worked example
Can a group of order 12 have a subgroup of order 5?
Lagrange: the subgroup's order has to divide 12
The divisors of 12 are 1, 2, 3, 4, 6 and 12
AnswerNo, because 5 is not on that list
Most common mistakeReading Lagrange's theorem backwards: assuming every divisor gives a subgroup, though the group A4 of order 12 has no subgroup of order 6.
You can map one group into another so the operation survives, then find the kernel and the image. You can also decide whether the map is an isomorphism.
Verify that a proposed map sends products to products.
Find the kernel as everything that lands on the identity.
Show that a trivial kernel makes the map one-to-one.
Worked example
Map each integer to its remainder mod 4. Find the kernel.
The kernel is every n whose remainder is 0
That is 0, 4, -4, 8, -8 and so on
AnswerThe kernel is the set of all multiples of 4
Most common mistakeCalling the kernel {0}: only the single integer 0 gets listed, though every other multiple of 4 also lands on the identity.
You can build a new group whose elements are whole cosets, once the subgroup is normal. You can also apply the first isomorphism theorem.
Check normality by conjugating the subgroup and getting it back.
List the elements of the integers mod 4 as four cosets.
Use the first isomorphism theorem to identify a quotient with an image.
Worked example
Describe the integers mod 4 as a quotient group
Take the group of integers and the subgroup of multiples of 4
The cosets are 0 + N, 1 + N, 2 + N and 3 + N
AnswerA group of order 4
Most common mistakeBuilding a quotient by a subgroup that is not normal: in S3 the two-element subgroup generated by a single swap is not normal, so its cosets do not multiply consistently.
You can divide, factor and test irreducibility for polynomials whose coefficients live in a ring or field. You can also apply Eisenstein's criterion.
Divide one polynomial by another and read off the remainder.
Apply Eisenstein's criterion with a chosen prime.
Explain why a polynomial can be irreducible over one field and not another.
Worked example
Is x squared + 1 irreducible over the reals? Over the complex numbers?
Over the reals it has no root, since x squared is never negative
Over the complex numbers it factors as (x + i)(x - i)
AnswerIrreducible over the reals, reducible over the complex numbers
Most common mistakeCalling a polynomial irreducible without naming the field: x squared - 2 is irreducible over the rationals but splits over the reals.
You can build a larger field by adding a root and measure how much larger it is with a degree. You can also multiply degrees through a chain with the tower rule.
Find the minimal polynomial of the square root of 2 over the rationals.
Compute the degree of that extension.
Use the tower rule to multiply degrees along a chain of fields.
Worked example
Find the degree of the rationals extended by the square root of 2
The square root of 2 is a root of x squared - 2, which does not factor over the rationals
A basis for the extension is 1 and the square root of 2
AnswerDegree 2
Most common mistakeReading the degree off the root symbol instead of the minimal polynomial: calling the cube root of 2 a degree-2 extension when x cubed - 2 makes it degree 3.
Before timing yourself, check whether you can explain Groups from a blank page. Then connect it to Subgroups. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Linear algebra calculator can test calculations and representations used in Abstract Algebra. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Abstract Algebra practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Groups if you are following the full sequence. If that unit feels automatic, open the Abstract Algebra practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Abstract Algebra test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Linear algebra calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
