Finding a nonempty subset that is not a subgroup

abstract-algebra

Let $G=\mathbb {R} \setminus \{0\}$ with multiplication and $H=(0,1]$


This set $H$ can lack inverses. Let $G = \mathbb Z$ and $H = \mathbb N_0$ (operation is addition).


Take the group $G$ as $\mathbb Z$ with addition and the subgroup $H$ as set of all even positive integers. $H$ is closed under addition but doesn't have inverse so $H$ is not a subgroup

Related

If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax+by=1$. [duplicate]
Find a basis of the $k$ vector space $k(x)$ [duplicate]
Summation of series involving binomial coefficients and polynomial of degree at most n-1
Variation on Vandermonde's identity
How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$?
Can $3p^4-3p^2+1$ be square number?
How to find the last digit of $3^{1000}$?
Proof that $a\mid b \land b\mid c \Rightarrow a\mid c $
If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$? [duplicate]
$R$ is an algebra over an infinite field. If $\exists$ ideals s.t. $J\subseteq \bigcup_{k=1}^nI_k$ then $J\subseteq I_k$ for some $k$
$k$ times's draw from $n$ numbers with replacement, each number at least appear once
Prove by Double Counting Method $\sum\limits_{k = 0}^m \binom{m}{k}\binom{n}{r + k} = \binom{m + n}{m + r}$ [duplicate]