let g be an element (non-identity) of a group G then g is its own inverse iff g is of order 2

An Abelian group is a group for which the elements commute (i.e., for all elements and. ). Abelian groups therefore correspond to groups with symmetric multiplication tables. All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal.
to show that (z,+) is an abelian group we need following properties
1) closure property
2) associative property
3) existence of identity
4) existence of inverse
5) commutative property
The reason why (Z, *) is not a group is that most of the elements do not have inverses. Furthermore, addition is commutative, so (Z, +) is an abelian group. The order of (Z, +) is infinite. The next set is the set of remainders modulo a positive integer n (Zn), i.e. {0, 1, 2, ..., n-1}.
show that cube root of unity form a group under multiplication. in abstract algebra ( G, *) is a group under multiplication where G= ( 1,w,w^2) when w^3= 1 abstract algebra
show that {2^n : n z } form a group under multiplication.
show that S = ( a + b√2 : a,b belong to Q ) form a group under multiplication
the set Un of n, nth root of unity forms an abelian group under multiplication.
prove that G=( 1, -1, i, -i ) form an abelian group under multiplication

1) cube root of unity is a group under multiplication, (G,*) is a group where G = ( 1, w, w^2).

2) show that S={2^n : n ∈ z } form a group under multiplication.

3) M2 be the set of all upper triangular 2 by 2 non singular matrice from a group under multiplication.

4) show that the set S = ( a+b√2: a,b ∈ Q ) form a group under multiplication.

5) the set S = ( xi + 0j : x ∈ R ) of vector in Euclidean plane form a group under addition.

6) Group theory, prove that the nth roots of unity forms an abelian group under multiplication.

7) Group theory, prove that the set of 2 by 2 matrices form an abelian group under multiplication.

8) Group theory, prove that G = ( 1, -1, i, -i ) forms an abelian group under multiplication.

9) let g be an element (non-identity) of a group G then g is its own inverse iff g is of order 2.

10) let g1g2 be two elements of a finite group G, then the order of g1g2 is the same as that of g2g1.

11) Group theory, prove that the order of an element g in G is equal to the order of its inverse.

12) Group theory, if every non- identity element of a group has order 2 then G is an abelian group.

13) Group theory, prove that every cyclic group is abelian.

14) Group theory, Every group of prime order is cyclic.

15) Group theory, two step subgroup test, two step test for subgroup.

16) Group theory, prove that Intersection of two subgroup of group G is subgroup.

17) Group theory, prove that Center of a group is subgroup of G.

18) Group theory, If H is a subgroup of G then aHa^-1 (conjugate of subgroup) is also a subgroup.

19) Group theory, If H and K are two subgroup of a group G whose order is relatively prime then H∩K=e.

20) Group theory, prove that any two right cosets are either disjoint or identical.

21) Group theory, state and prove Lagrange's theorem // Lagrange's theorem.

22) Let H be a subgroup of a group G then quotient set 𝐆/𝐇={𝒈𝑯 :𝒈∈𝑮} form a group under multiplication.

23) Isomorphism ll Show that mapping 𝝍:(𝑹×𝑹,+)→{𝑽_𝟐 (𝑹),+} define by 𝝍(𝒙,𝒚)=𝒙𝒊 +𝒚𝒋 is an isomorphism.

24) Group theory, prove that Every infinite cyclic group is isomorphic to group ( z, +) of all integers.

25) Group theory, prove that the order of an element of a group divides the order of group G.

26) let G be a group & x be an element of odd order in G then for any element y in G their exist y^2=x.