Help with proof regarding normal subgroups and cosets

In summary, the conversation discusses the concept of normal subgroups in a group. It is stated that a subgroup H is normal in G if every left coset of H is equal to some right coset of H. The conversation also includes a proof of this statement, showing both the "=>" and "<=" directions. A warning is also given about using set multiplication in proofs.
  • #1
ssayani87
10
0

Homework Statement



I think I've got this one about figured out, I just wanted someone to check it over. (For this problem, (a-1) is a inverse, (b-1) b inverse, etc.)

"Let G be a group, H a subgroup of G.

Then, H is normal in G iff every left coset of H is equal to some right coset of H"

Homework Equations



H normal in G implies aH(a-1) = H for all a in G

The Attempt at a Solution



Let a,b be elts of G, and H a normal sbgp of G.

(=>):

By normal condition, aH(a-1) = H for all a in G

-> aH = Ha (right multiply by a)

H normal in G implies the left coset of H by a coincides with the right coset of H by a.

Implies that if H is normal, every left coset of H is equal to some right coset of H.

(<=):

Let the left coset of H by a be equal to some right coset of H by b.

So, aH = Hb

-> aH(b-1) = H (right multiply by (b-1) )

H = aH(b-1) implies (by taking inverses) that H = bH(a-1)

Finally, H = HH = (aH(b-1))(bH(a-1)) = aH((b-1)b)H(a-1) = aH(e)H(a-1) = aHH(a-1) = aH(a-1)

Then we have H = aH(a-1)

Similar argument shows that H = bH(b-1)

So, H is normal.

In conclusion, if every left coset of H is equal to some right coset of H, then H is normal in G.

I think that handles both directions (<=) & (=>).

To me, this seems like a pretty pertinent question regarding normal sbgps so I'd like to make sure I hit it on the head.

Thanks!
 
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  • #2
you can use the "SUP" tags to produce inverses, like so:

x-1

(it's the icon under "advanced" that looks like x2).

your "=>" part is fine, you exhibit some right coset that aH is equal to, for every a in G, namely Ha.

for your "<=" part, note that while a is arbitrary, b in fact depends on a, that is, given aH, b is some element such that aH = Hb (it might be the case that there is only one such b, for example).

so you only need to show that aHa-1 = H, because this will be true for ANY a in G, whereas bHb-1 = H only holds for those b for which aH = Hb (for a specific a).

that is, the subsequent proof that bHb-1 = H is unecessary, and considerably less general than aHa-1 = H (we have no prior conditions on a, but we do for b).

a general warning on "set multiplication": be aware that you cannot always use facts about AB to prove facts about ab. for example, if G is abelian, AB = BA, but it is often the case that AB = BA, but G is not abelian. so it's a good idea to check that your statements about sets like:

(aHb-1)(bHa-1) = H still make sense at "the element level":

(ahb-1)(bh'a-1) should yield an element of H.

being able to treat certain subsets of G (cosets of H) like elements, is a special property of normal subgroups, and statements like HH = H, only apply to certain subsets of G. what you have written is indeed true, just be aware that thinking of sets as things you can multiply, CAN get you into trouble (you can't always use the "group rules" of G to calculate these).
 

What is a normal subgroup?

A normal subgroup is a subgroup of a group that is invariant under conjugation by elements of the larger group. This means that if an element of the larger group is multiplied on the left and the right by an element of the normal subgroup, the result will still be an element of the normal subgroup.

How do I prove that a subgroup is normal?

To prove that a subgroup is normal, you must show that for every element in the larger group, its conjugates with elements of the subgroup are also in the subgroup. This can be done by directly checking each conjugate or by using the definition of normality and the properties of conjugation.

What is a coset?

A coset is a subset of a group that is formed by multiplying all elements of a subgroup by a single element of the larger group. It is denoted by gH, where g is the chosen element and H is the subgroup. Cosets help to partition a group into smaller, equal-sized subsets.

How do I show that two cosets are equal?

To show that two cosets are equal, you must prove that every element in one coset is also in the other coset, and vice versa. This can be done by showing that the two cosets are subsets of each other, or by showing that their intersection is equal to both cosets.

How are normal subgroups and cosets related?

A normal subgroup is a subgroup that is invariant under conjugation by all elements of the larger group. This means that the left and right cosets of the subgroup are equal. Additionally, the set of all cosets of a normal subgroup forms a group, called the quotient group, which is isomorphic to the larger group divided by the normal subgroup.

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